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Permanent Address: Addis Ababa University, Faculty of Technology, Addis Ababa, Ethiopia

Simulation Model for Pipeline Network System with Non-pipe

Elements for Natural Gas Transmission

AP. Ir. Dr. Mohd Amin Abd Majid

UNIVERSITI TEKNOLOGI PETRONAS

SIMULATION MODEL FOR PIPELINE NETWORK SYSTEM WITH

NON-PIPE ELEMENTS FOR NATURAL GAS TRANSMISSION

by

ABRAHAM DEBEBE WOLDEYOHANNES

The undersigned certify that they have read, and recommend to the Postgraduate Studies Programme for acceptance this thesis for the fulfilment of the requirements for the degree stated.

Signature: ____________________________________ Main Supervisor: Assoc. Prof. Ir. Dr. Mohd Amin Abd Majid

Signature: ______________________________________ Head of Department: Dr. Ahmad Majdi Bin Abdul Rani

Date: ______________________________________

SIMULATION MODEL FOR PIPELINE NETWORK SYSTEM WITH

NON-PIPE ELEMENTS FOR NATURAL GAS TRANSMISSION

by

ABRAHAM DEBEBE WOLDEYOHANNES

A Thesis

Submitted to the Postgraduate Studies Programme

as a Requirement for the Degree of

DOCTOR OF PHILOSOPHY

MECHANICAL ENGINEERING

UNIVERSITI TEKNOLOGI PETRONAS

BANDAR SERI ISKANDAR,

PERAK

MAY 2010

iv

DECLARATION OF THESIS

Title of thesis Simulation Model for Pipeline Network System with Non-pipe Elements

for Natural Gas Transmission

I , ABRAHAM DEBEBE WOLDEYOHANNES, hereby declare that the thesis is based on

my original work except for quotations and citations which have been duly acknowledged. I

also declare that it has not been previously or concurrently submitted for any other degree at

UTP or other institutions.

Witnessed by ________________________________ __________________________ Signature of Author Signature of Supervisor Permanent address: Addis Ababa University, AP. Ir. Dr. Mohd Amin Abd Majid Faculty of Technology, Addis Ababa, Ethiopia

Date : _____________________ Date : ________________

v

DEDICATION

To my beloved wife, Meaza Feyssa, and my daughter, Halleluiah Abraham,

for being with me during this time

vi

ACKNOWLEDGEMENTS

First and for most, thanks to my GOD for giving me the patience to start and

finalize this thesis work. I would like to extend my sincere appreciation to my

supervisor, Assoc. Prof. Ir. Dr. Mohd Amin Abd Majid, for his valuable support and

guidance throughout the completion of this thesis. He spent many of his precious

time to teach and guide me. His encouragement, creative suggestions, critical

comments and everlasting enthusiastic support have greatly contributed to this thesis.

I would also like to thank my beloved wife, Meaza Feyssa Tolla, for her love and

support during all the time. Not only has she encouraged me to pursue my academic

interests, she also sacrificed a lot to be with me during this time. For this, I will be

forever grateful and acknowledge with gratitude and love the importance of her in

my life. Furthermore, great thanks to my daughter Halleluiah for bringing me many

funs and pleasures in my life.

I would also like to thank the Universiti Teknologi PETRONAS (UTP) for

providing me grant and facility for the research. Furthermore, great thanks for Addis

Ababa University for granting my study leave to continue with my graduate studies. I

would also like to express my gratitude to PETRONAS GAS BERHAD for the

information and data about the gas pipeline networks. Great thanks to Dr. Dereje

Engida for revising the thesis and forwarding his comments. It is my pleasure to

acknowledge the support from Mr Noraz Al-Kahair Bin Noran for translating the

abstract to Bahasa Melayu.

Last but not least, great thanks to my fellow friends for providing me a good

atmosphere during my stay at UTP. There are also many people at UTP in one way

or another who have helped me. Thank you very much.

vii

ABSTRACT

The ever growing demand for natural gas enhances the development of complex

transmission pipeline network system (TPNS) which requires simulation procedures

for design and operation of the network. TPNS simulation is usually performed in

order to determine the nodal pressures, temperatures and flow variables under

various configurations. These variables are essential for analyzing the performance

of the TPNS. The addition of non-pipe elements like compressor stations, valves,

regulators and others make TPNS simulation analysis more difficult.

A new simulation model was developed based on performance characteristic of

compressors and the principles of conservation of energy and mass of the system to

analyze TPNS with non-pipe elements for various configurations. The TPNS

simulation model analyzes single phase gas flow and two-phase gas-liquid flow. The

simulation model also takes into account temperature variation and age of the pipes.

The model was designed for the evaluation of the unknown pressure, flow and

temperature variables for the given pipeline network. The two solution schemes

developed were iterative successive substitution scheme for simple network

configurations and a generalized solution scheme which adopt Newton-Raphson

algorithm for complex network configurations. The generalized Newton-Raphson

based TPNS simulation model was tested based on the three most commonly found

network configurations, namely: gunbarrel, branched and looped pipeline. In all the

tests conducted, the solutions to the unknown variables were obtained with a wide

range of initial estimations. A maximum of ten iterations were required to get

solutions to nodal pressure, flow and temperature variables with relative percentage

errors of less than 10-11.

viii

The results of TPNS simulation model were compared with Newton loop-node

method based on looped pipeline network configurations and an exhaustive

optimization technique based on gunbarrel pipeline network configuration. For both

cases, the results indicate that the model is able to provide solutions similar to the

compared models. In addition, the TPNS simulation model provides detail

information for the compressor stations. This information is essential for evaluation

of the performance of the system.

The application of the TPNS simulation model for real pipeline network system

was also conducted based on existing pipeline network system. Three modules of

TPNS simulation model which included input parameter analysis, function

evaluation and network evaluation module were evaluated using the data taken from

the real system. Analyses of the performance of compressor for existing pipeline

network system which included discharge pressure, compression ratio and power

consumption were also conducted using the developed TPNS simulation model. The

performance characteristics maps generated by the developed TPNS simulation

model show the variation of discharge pressure, compression ratio, and power

consumption with flow rate similar to the one available in the literatures.

Based on the results from the simulation tests and validation of the model, it is

noted that the developed TPNS simulation model could be used for performance

analysis to assist in the design and operations of transmission pipeline network

systems.

ix

ABSTRAK

Permintaan yang semakin tinggi untuk gas asli memerlukan pembangunan

kompleks sistem penghantaran saluran paip berangkaian (transmission pipeline

network system; TPNS) ditingkatkan di mana ia memerlukan prosedur-prosedur

simulasi untuk rekaan dan operasi rangkaian. Simulasi TPNS biasanya digunakan

bagi tujuan untuk menentukan pelbagai tekanan, suhu dan pembolehubah aliran

berbuku di bawah pelbagai konfigurasi. Pembolehubah ini adalah penting untuk

mengkaji prestasi TPNS. Penambahan elemen selain paip seperti stesen pemampat,

injap, pengatur dan lain-lain menyebabkan analisis simulasi TPNS lebih sukar.

Model simulasi baru dimajukan berasaskan kepada prestasi mesin pemampat dan

prinsip penjimatan tenaga dan juga jisim pada sistem bagi menganalisis TPNS

dengan unsur selain paip untuk pelbagai konfigurasi. Model simulasi TPNS

menganalisis aliran gas satu-fasa dan aliran cecair gas dua-fasa. Model simulasi ini

turut mengambil kira pelbagai suhu dan usia paip itu.

Model ini direka untuk menilai pembolehubah tekanan, aliran dan suhu yang

tidak diketahui untuk rangkaian saluran paip. Dua skim penyelesaian yang dirangka

adalah skim lelaran penggantian berturut-turut untuk konfigurasi rangkaian mudah

dan skim penyelesaian am yang menggunakan algoritma Newton-Raphson untuk

konfigurasi rangkaian yang sukar. Model simulasi am Newton-Raphson berasaskan

TPNS diuji berdasarkan tiga rangkaian paling umum yang didapati di konfigurasi

rangkaian iaitu: laras, bercabang dan lingkaran paip. Dalam semua ujian yang

dijalankan, penyelesaian pada pembolehubah yang tidak dikenali itu diperolehi

dengan satu anggaran julat awal yang luas. Iterasi yang diperlukan adalah sepuluh

lelaran untuk mendapatkan penyelesaian untuk pembolehubah tekanan, aliran dan

suhu berbuku dengan peratusan kesilapan relatif kurang daripada 10-11.

Keputusan model simulasi TPNS dibandingkan dengan kaedah gelung nod

Newton berdasarkan konfigurasi menggelung rangkaian saluran paip dan teknik

pengoptimuman yang lengkap berdasarkan konfigurasi laras rangkaian saluran paip.

Untuk kedua-dua kes, keputusan menunjukkan bahawa model ini boleh menyediakan

x

penyelesaian yang mirip dengan model yang sebanding dengannya. Seperkara lagi,

model simulasi TPNS menyediakan maklumat terperinci untuk stesen pemampat.

Maklumat ini adalah penting untuk penilaian prestasi sistem.

Penggunaan model simulasi TPNS juga diterapkan untuk sistem rangkaian

saluran paip sebenar berdasarkan saluran paip sistem rangkaian yang sedia ada. Tiga

modul bagi model simulasi TPNS ini termasuklah analisis parameter input, penilaian

fungsi dan penilaian rangkaian modul yang dinilai menggunakan data yang diambil

dari sistem sebenar. Analisis prestasi pemampat untuk sistem rangkaian saluran paip

sedia ada termasuklah tekanan luahan, nisbah mampatan dan penggunaan kuasa juga

dikendalikan dengan menggunakan model simulasi TPNS yang dibangun kan. Ciri

prestasi ‘peta’ dihasilkan oleh model simulasi TPNS yang dibangun kan

menunjukkan pelbagai tekanan luahan, nisbah mampatan, dan penggunaan kuasa

dengan kadar aliran yang menyerupai dengan hasil kajian orang lain.

Berdasarkan keputusan dari simulasi dan pengesahan model ini, ia menunjukkan

model simulasi TPNS yang dibangun kan boleh digunakan untuk menganalisis

prestasi bagi memudahkan reka bentuk dan pengendalian sistem penghantaran

saluran paip berangkaian.

xi

COPYRIGHT PAGE

In compliance with the terms of the Copyright Act 1987 and the IP Policy of the university, the copyright of this thesis has been reassigned by the author to the legal entity of the university,

Institute of Technology PETRONAS Sdn Bhd.

Due acknowledgement shall always be made of the use of any material contained in, or derived from, this thesis.

© Abraham Debebe Woldeyohannes, 2010

Institute of Technology PETRONAS Sdn Bhd All rights reserved.

xii

TABLE OF CONTENTS

DECLARATION OF THESIS ................................................................................. iv

DEDICATION ........................................................................................................... v

ACKNOWLEDGEMENTS ..................................................................................... vi

ABSTRACT ............................................................................................................ vii

ABSTRAK ............................................................................................................... ix

COPYRIGHT PAGE ................................................................................................ xi

TABLE OF CONTENTS ........................................................................................ xii

LIST OF TABLES ................................................................................................. xvi

LIST OF FIGURES .............................................................................................. xviii

NOMENCLATURES ........................................................................................... xxiii

PREFACE ........................................................................................................... xxvii

Chapter

1. INTRODUCTION .................................................................................................. 1

1.1 Background .................................................................................................. 1

1.2 Problem Statement ....................................................................................... 4

1.3 Research Objective ...................................................................................... 6

1.4 Research Scopes .......................................................................................... 7

1.5 Structure of Thesis ....................................................................................... 8

2. LITERATURE REVIEW .................................................................................... 10

2.1 Introduction ............................................................................................... 10

2.2 Optimization of Gas Pipeline Networks .................................................... 11

xiii

2.2.1 Dynamic Programming ...................................................................... 12

2.2.2 Mathematical Programming ............................................................... 15

2.2.3 Expert System .................................................................................... 17

2.2.4 Analytical Hierarchical and Network Reduction ............................... 19

2.3 Simulation of Gas Pipeline Networks ....................................................... 21

2.3.1 Simulation of Gas Networks without Non-pipe Elements ................. 22

2.3.2 Simulation of Gas Networks with Non-pipe Elements ...................... 26

2.4 Simulation Models with Two-phase, Temperature and Corrosion ........... 30

2.4.1 Two-phase Flow Models .................................................................... 31

2.4.2 Simulation Model with Temperature Variations ............................... 37

2.4.3 Internal Corrosion .............................................................................. 39

2.5 Solution Principles Applied in Gas Pipeline Network Simulations .......... 41

2.6 Summary.................................................................................................... 42

3. METHODOLOGY .............................................................................................. 45

3.1 Introduction ............................................................................................... 45

3.2 Mathematical Formulation of the Simulation Model ................................ 47

3.2.1 Single Phase Flow Equations Modeling ............................................ 47

3.2.2 The Looping Conditions .................................................................... 50

3.2.3 Compressor Characteristics Equations ............................................... 52

3.2.4 Mass Balance Equations .................................................................... 60

3.3 Successive Substitution based TPNS Simulation Model .......................... 61

3.3.1 Case 1A: Single Compressor Station and Two Customers Module .. 66

3.3.2 Case 2A: Two Compressor Stations and Two Customers Module .... 69

xiv

3.4 Newton-Raphson based TPNS Simulation Model .................................... 72

3.4.1 Case 1B: Gunbarrel Pipeline Network Module .................................. 76

3.4.2 Case 2B: Branched Pipeline Network Module .................................. 79

3.4.3 Case 3B: Looped Pipeline Network Module ...................................... 81

3.4.4 Case Study: Malaysia Gas Transmission System ............................. 85

3.5 Enhanced Newton-Raphson based TPNS Simulation Model .................... 88

3.5.1 Two-phase Gas-liquid Flow Modeling .............................................. 88

3.5.2 The Effect of Internal Corrosion ........................................................ 96

3.5.3 Modeling Temperature Variations ................................................... 101

3.6 Performance Evaluation of the TPNS Simulation Model ....................... 103

3.7 Summary .................................................................................................. 104

4. RESULTS AND DISCUSSIONS ..................................................................... 107

4.1 Introduction ............................................................................................. 107

4.2 Successive Substitution based TPNS Simulation Model ........................ 107

4.2.1 Case 1A: Single Compressor Station and Two Customers Module . 108

4.2.2 Case 2A: Two Compressor Stations and Two Customers Module .. 110

4.2.3 Limitations of Successive Substitution TPNS Simulation Model ... 112

4.3 Newton-Raphson based TPNS Simulation Model .................................. 114

4.3.1 Case 1B: Gunbarrel Pipeline Network Module ................................ 115

4.3.2 Case 2B: Branched Pipeline Network Module ................................ 120

4.3.3 Case 3B: Looped Pipeline Network Module .................................... 123

4.3.4 The Malaysia Gas Transmission Network Case Study .................... 132

4.3.5 The Case of Divergent in TPNS Simulation Model ......................... 143

xv

4.4 Enhanced Newton-Raphson based TPNS Simulation Model ................. 144

4.4.1 Two-phase Gas-liquid Flow Analysis .............................................. 144

4.4.2 The Effect of Internal Corrosion ...................................................... 150

4.4.3 Non-isothermal Flow Analysis ........................................................ 161

4.5 Verification and Validation of the TPNS Simulation Model .................. 163

4.5.1 TPNS Conceptual Model Validation and Verification .................... 165

4.5.2 Operational Validation ..................................................................... 167

4.6 Summary.................................................................................................. 177

5. CONCLUSIONS AND RECOMMENDATIONS ........................................... 179

5.1 Conclusions ............................................................................................. 179

5.2 Contributions of the Research ................................................................. 181

5.3 Recommendations ................................................................................... 183

REFERENCES ....................................................................................................... 185

APPENDICES

A. Snapshot of the Developed TPNS Simulation Code using Visual C++

B. Results of TPNS Simulation Model for Gunbarrel Network

C. Successive Substitution based Simulation Results (Case 1A)

D. Divergent Case in TPNS Simulation (Case 2C)

xvi

LIST OF TABLES

Table 2.1 Gas flow equations and range of applications adapted from [7] and [23]

................................................................................................................................ 25

Table 2.2 Summary of TPNS Simulation Models .................................................. 43

Table 3.1 Single phase flow equations and functional representations .................. 49

Table 3.2 Values of coefficients for compressor equations .................................... 56

Table 3.3 Number of equations and unknowns for TPNS configurations .............. 62

Table 3.4 Gas properties ......................................................................................... 63

Table 3.5 Summary of flow equations and their corresponding functional

representations for the given TPNS ........................................................................ 67

Table 3.6 Pressure data for single CS and two customers ...................................... 68

Table 3.7 Pipe data for single CS and two customers ............................................ 68

Table 3.8. Summary of flow equations and their corresponding functional

representations for the given TPNS ........................................................................ 70

Table 3.9 Summary of compressor equations and functional representations ....... 70

Table 3.10 Pipe data for two CSs and two customers ............................................ 72

Table 3.11 Summary of flow equations for the given gunbarrel TPNS ................. 77

Table 3.12 Summary of compressor equations and functional representations ..... 78

Table 3.13 Pressure requirements for two CSs and single customer ...................... 78

Table 3.14 Pipe data for two CSs and single customer .......................................... 78

Table 3.15 Summary of flow equations and their functional representation .......... 80

Table 3.16 Summary of mass balance equations and functional representations ... 80

Table 3.17 Pipe data for single CS and five customers .......................................... 81

xvii

Table 3.18 Summary of flow equations and functional representations ................ 83

Table 3.19 Summary of mass balance equations and functional representations .. 84

Table 3.20 Pipe data for single CS and eight customers ........................................ 85

Table 3.21 Two-phase flow equations and functional representations .................. 93


Table 3.23 Summary of flow equations and their functional representation ......... 95


Table 3.25 Summary of flow equations and their functional representation

considering corrosion ........................................................................................... 100

Table 3.26 Summary of temperature equations for gunbarrel TPNS ................... 103

Table 3.27 Various cases considered for the analysis using developed TPNS

simulation model .................................................................................................. 106

Table 4.1 Results of nodal pressures and flow variables (case 1B) ..................... 115

Table 4.2 Results of nodal pressures [47] ............................................................ 118

Table 4.3 Results of nodal pressures after ten iterations ...................................... 124

Table 4.4 Results of main and branch flow variables after 10 iterations ............. 125

Table 4.5 Pipe data [7] .......................................................................................... 128

Table 4.6 Pressure and flow requirements [7] ...................................................... 129

Table 4.7 Results of input analysis for pipes ........................................................ 134

Table 4.8 Results of nodal pressures and flow variables after 10 iterations ........ 136

Table 4.9 Comparison of nodal pressures for different pipe ages ........................ 153

Table 4.10 Variation of flow with ages of pipes .................................................. 160

Table 4.11 Results of temperature variables for the first 10 iterations ................ 161

xviii

LIST OF FIGURES

Figure 2.1 Gunbarel structure transmission pipeline network adapted from [8] .... 13

Figure 2.2 Flow of the gas though pipe adapted from [7] ...................................... 24

Figure 2.3 Typical performance map of centrifugal compressor [59] .................... 29

Figure 2.4 Flow pattern in horizontal and near-horizontal pipes [70] .................... 34

Figure 2.5 Schematic of homogeneous no-slip flow model adapted from [70] ..... 36

Figure 2.6 Analysis of temperature variations adapted from [23] .......................... 38

Figure 2.7 Effect of years in service on pipe roughness [22] ................................ 40

Figure 3.1 Structure of the simulation model for transmission pipeline networks . 46

Figure 3.2 Pipe joining two consecutive nodes ...................................................... 48

Figure 3.3 Part of transmission pipeline network system ....................................... 49

Figure 3.4 Looped pipeline network system........................................................... 51

Figure 3.5 Normalized head and normalized flow rate for different speeds of a

typical centrifugal compressor ................................................................................ 53

Figure 3.6 Efficiency and normalized flow rate for different speeds of a typical

centrifugal compressor............................................................................................ 54

Figure 3.7 Compressors operating in series within a station .................................. 56

Figure 3.8 Compressors working in parallel within a station ................................ 58

Figure 3.9 Comparison of selected data and approximated data based on the

compressor data from [59] ...................................................................................... 59

Figure 3.10 Comparison of calculated and actual pressure ratios ......................... 60

Figure 3.11 Mass balance formulation ................................................................... 61

Figure 3.12 Possible information flow blocks for compressor .............................. 64

xix

Figure 3.13 The developed flow chart for successive substitution based TPNS

simulation model .................................................................................................... 65

Figure 3.14 Gas transmission system for two customers ....................................... 66

Figure 3.15 IFD of pipeline network system with two customers ........................ 68

Figure 3.16 Pipeline network with two CSs and two customers ............................ 69

Figure 3.17 IFD for pipeline network with two CSs .............................................. 71

Figure 3.18 Flow chart of the TPNS simulation model based on Newton-Raphson

scheme .................................................................................................................... 75

Figure 3.19 Gunbarrel pipeline network system with two CSs .............................. 77

Figure 3.20 Branched pipeline network system ..................................................... 79

Figure 3.21 Looped pipeline network system ........................................................ 82

Figure 3.22 Part of the existing pipeline network system ...................................... 88

Figure 3.23 Proposed scheme for incorporating the age of the pipes in flow

equations ................................................................................................................. 97

Figure 3.24 The effect of length of service of the pipe on friction factor .............. 98

Figure 4.1 Convergence of pressure variables (case 1A) ..................................... 109

Figure 4.2 The convergence of flow variables (Case 1A) .................................... 110

Figure 4.3 The convergence of pressure variables (case 2A) ............................... 111

Figure 4.4 The convergence of flow variables (case 2A) ..................................... 112

Figure 4.5 Pipeline network with three CSs serving four customers ................... 113

Figure 4.6 IFD for pipeline network with three CSs and four customers ............ 114

Figure 4.7 Convergence of pressure variables (case 1B) ..................................... 117

Figure 4.8 Convergence of flow variable (case 1B) ............................................ 117

Figure 4.9 Comparison of nodal pressures ........................................................... 120

xx

Figure 4.10 Convergence of pressure (case 2B) ................................................... 122

Figure 4.11 Convergence of main flow variables (case 2B) ................................ 122

Figure 4.12 Convergence of lateral flow variables (case 2B) .............................. 123

Figure 4.13 Convergence of P1 for the first ten iterations for the initial estimations

(case 3B) ............................................................................................................... 126

Figure 4.14 Convergence of P2 for the first ten iterations for the initial estimation

(case 3B) ............................................................................................................... 126

Figure 4.15 Convergence of Q1 for the first ten iterations for the initial estimations

(case 3B) ............................................................................................................... 127

Figure 4.16 Pipeline networks with 10 pipes and two compressor stations [7] ... 128

Figure 4.17 Comparison of flow rates between TPNS simulation and results of

Osiadacz [7] .......................................................................................................... 131

Figure 4.18 Comparison of nodal pressures between TPNS simulation and results

of Osiadacz [7] ...................................................................................................... 132


.............................................................................................................................. 137


.............................................................................................................................. 138


.............................................................................................................................. 139

Figure 4.22 Convergence of the main flow variables ........................................... 140

Figure 4.23 Discharge pressure variations with flow for various speeds ............. 141

Figure 4.24 Compression ratio variations with flow for various speeds .............. 142

Figure 4.25 Variation of power consumption with flow for various speed .......... 143

Figure 4.26 Convergence of pressure variables (Case 1C) ................................... 145

xxi

Figure 4.27 Convergence of flow variable (Case 1C) .......................................... 145

Figure 4.28 Convergence of pressure variables (Case 2C) .................................. 147

Figure 4.29 Convergence of main flow variables (Case 2C) ............................... 148

Figure 4.30 Convergence of lateral flow variables (Case 2C) ............................. 148

Figure 4.31 Effect of liquid holdup on nodal pressures ....................................... 149

Figure 4.32 Effect of liquid holdup on flow through main pipes ......................... 150

Figure 4.33 Convergence of pressure variables with corrosion (Case 3C) .......... 151

Figure 4.34 Convergence of flow variable with corrosion (Case 3C) .................. 152

Figure 4.35 Variation of flow with age of the pipe .............................................. 154

Figure 4.36 Variation of compression ratio with age of the pipe ......................... 155

Figure 4.37 Variation of the power consumption with age of the pipes .............. 156

Figure 4.38 Convergence of pressure variables (Case 4C) .................................. 157

Figure 4.39 Convergence of flow through main pipes (Case 4C) ........................ 157

Figure 4.40 Convergence of flow through lateral pipes (Case 4C) ...................... 158

Figure 4.41 Nodal pressures for different ages of pipes ....................................... 159

Figure 4.42 Flow through pipes for different ages of pipes ................................. 160

Figure 4.43 Convergence of pressures for non-isothermal conditions ................. 162

Figure 4.44 Convergence of flow variables under non-isothermal conditions .... 162

Figure 4.45 Convergence of some of the temperature variables .......................... 163

Figure 4.46 Simplified version of the modeling process [95] .............................. 165

Figure 4.47 Variations of flow rate with number of compressors ........................ 170

Figure 4.48 Variation of compression ratio with number of compressors ........... 171

Figure 4.49 Variation of power consumption with number of compressors ........ 172

xxii

Figure 4.50 Variation of main flow rate with source pressure ............................. 174

Figure 4.51 Variation of compression ratio with source pressure ........................ 175

Figure 4.52 Variation of power consumption with source pressure ..................... 176

xxiii

NOMENCLATURES

41 AA − Constants for normalized compressor head equations

PA Cross-sectional area of pipe

AGA American gas association

41 BB − Constants for normalized compressor efficiency equations

BHP Break horse power

BSCFD Billion standard cubic feet per day

C1, C2 Constants used to name pipes

PC Specific heat at constant pressure

CR Compression ratio

CS Compressor station

vC Specific heat at constant volume

D Diameter

iD Customer located at i

ijD Diameter of pipe joining node i and j

lD Volumetric flow rate of load pipe l

DP Dynamic programming

E Pipeline efficiency

f Darcy’s friction factor 'f Fanning friction factor

F Transmission factor

g Acceleration due to gravity

G Gas gravity

GA Genetic algorithm

GRG Generalized reduced gradient

h Elevation height

xxiv

H Adiabatic head

LH Liquid holdup

IFD Information flow diagram

k Specific heat ratio

ijK Pipe flow resistance between pipe node i and j

L Length

ijL Length of pipe joining node i and j

LNG Liquefied natural gas

M Mass flow rate

fM Total mass flux

GM Mass flow rate for the gas

LM Mass flow rate for the liquid

MINLP Mixed integer non linear programming

MMSCMD Million metric standard cubic meters per day

MMSCFD Million standard cubic feet per day

n Rotational speed of the compressor, exponent

cn Number of compressors

NDP Non sequential dynamic programming

jn Number of junctions within the pipeline network system

ln Number of loops within the pipeline network system

NLP Non linear programming

NP Number of unknown pressure variables

NQ Number of unknown flow variables

sn Number of compressor stations within the pipeline network system

NT Number of unknown temperature variables

NTotal Total number of unknown variables

P Pressure

avP Average pressure

xxv

sP , dP Suction, discharge pressure of the compressor

DP Demand pressure

nP Standard pressure condition

jq Volumetric flow rate of outgoing pipe j from the node

Q Volumetric flow rate

CiQ Gas flow rate to customer i

iQ Volumetric flow rate through pipe i

mQ Volumetric flow rate of the mixture

nQ Volumetric flow at standard conditions

r Roughness of the pipe as function of age

R Gas constant

airR Gas constant for air

Re Reynolds number

mRe Mixture Reynolds’s number

SA Simulated annealing

SS Successive substitution

PS Perimeter of the pipe

SCADA Supervisory control and data acquisition

t Number of incoming pipe to a node

T Temperature of gas

sT Average soil temperature

ST Temperature of the gas at the suction side of the compressor

ijT Temperature constant between upper node i and down node j

nT Standard temperature conditions

TPNS Transmission pipeline network system

u Number of outgoing pipes from a node

U Overall heat transfer coefficient

xxvi

Gv Velocity of the gas

Lv Velocity of the liquid

mv Velocity of the mixture

w Number of load pipes from a node

x Mass fraction (quality) X Vector representing unknown variables y Age of the pipe

Z Gas compressibility

1Z , 2Z Suction, discharge side compressibility of gas

Greek letters

α Void fraction

η Efficiency

Lλ No-slip liquid holdup

Gμ Viscosity of the gas

Lμ Viscosity of the liquid

mμ Viscosity of the mixtures

Gυ , Lυ specific gravity of the gas, liquid

θ Inclination angle

ρ Density

Gρ Density of the gas

Lρ Density of the liquid

NSρ The no-slip density of the mixture

mρ Density of the mixtures

wτ Shear stress at the wall of the pipe

xxvii

PREFACE

This thesis is submitted as a requirement for the degree of Doctor of Philosophy

in Mechanical Engineering for the author. It consists of the research work conducted

in the area of simulation of natural gas transmission from July 2006 to March 2010.

The problem was started from the observation of the congestions that usually

happened at natural gas distribution centers. The revision of the various literatures on

natural gas transportation showed that the distribution centers are the last system and

involved less volume of gas. It was observed that the problems occurred on the

transmission part of the pipeline network system. The main issues in transmission

pipeline network system (TPNS) were the determination of the nodal pressures,

flows and temperature variables which were used to evaluate the performance of the

system.

A simulation model was developed based on performance characteristic of

compressors and the principles of conservation of energy and mass of the system to

analyze TPNS with non-pipe elements for various configurations. The TPNS

simulation model analyzes single phase gas flow and two-phase gas-liquid flow. The

simulation model also takes into account temperature variation and age of the pipes.

The model was designed for the evaluation of the unknown pressure, flow and

temperature variables for the given pipeline network. The two solution schemes

developed were iterative successive substitution scheme for simple network

configurations and a generalized solution scheme which adopt Newton-Raphson

algorithm for complex network configurations. The results of TPNS simulation

model were compared with Newton loop-node method based on looped pipeline

network configurations and an exhaustive optimization technique based on gunbarrel

pipeline network configuration. For both cases, the results indicated that the model

is able to provide solutions similar to the compared models.

xxviii

Based on the results from the simulation tests and validation of the model, it is

noted that the developed TPNS simulation model could be used for performance

analysis to assist in the design and operations of transmission pipeline network

systems.

1

CHAPTER 1

INTRODUCTION

1.1 Background

Natural gas is becoming one of the most widely used sources of energy in the

world due to its environmental friendly characteristics. Usually, the location of

natural gas resources and the place where the gas is needed for various applications

are far apart. As a result, the gas has to be moved from deposit and production sites

to consumers either by trucks in the form of liquefied natural gas (LNG) or through

pipeline network systems. As reported in [1], short distances gas transportation by

pipelines is more economical than LNG transportation. The LNG transportation

incurs liquefaction costs irrespective of the distance over which it is moved. As a

result, the development of transmission pipeline network system (TPNS) for natural

gas is a key issue in order to satisfy the ever growing demand from the various

customers.

When the gas moves by using the TPNS, the gas flows through pipes and various

devices such as regulators, valves, and compressors. The pressure of the gas is

reduced mainly due to friction with the wall of the pipe and heat transfer between the

gas and the surroundings. Compressor stations are usually installed to boost the

pressure of the gas and keep the gas moving to the required destinations. It is

estimated that 3 to 5% of the gas transported is consumed by the compressors in

order to compensate for the lost pressure of the gas [2]-[4]. This is actually a huge

amount of gas especially for the network transmitting large volume of gas. At the

current price, this represents a significant amount of cost for the nation operating

2

large pipeline network system. For instance, considering the U.S. TPNS, Wu [2]

indicated that a 1% improvement on the performance of the transmission pipeline

network system could result a saving of 48.6 million dollars. Carter [3] also

presented that the cost of natural gas burned to power the transportation of the

remaining gas for the year 1998 is equivalent to roughly 2 billion dollars for U.S.

transmission system.

Investigation on various TPNS indicated that the overall operating cost of the

system is highly dependent upon the operating cost of the compressor stations which

represents between 25% and 50% of the total company’s operating budget as

discussed in [5] and [6]. Hence, compressor station is considered as one of the basic

elements in TPNS.

The main issues associated with both design and operating TPNS are minimizing

the energy consumption and maximizing the flow rate through pipes. Over the years,

numerical simulations of TPNS have been carried out in order to determine the

optimal operational parameters for given networks with various degrees of success

[2], [3], [7]-[9]. From the optimization perspective, the problem of developing an

optimal TPNS is nonlinear programming problem where the objective function is

typically nonlinear and non-convex, and some of the constraints are also nonlinear.

Different techniques are proposed in order to get the optimal parameters of TPNS by

either modifying the objective function or relaxing some of the constraints [10]-[15].

However, due to the complexity of the objective function and the constraints, the

determination of optimal parameters for TPNS is yet challenging from the

optimization perspective.

On the other hand, simulation has contributed significant achievements in

analyzing the TPNS problems [7], [9], [14], [16]- [20]. TPNS simulation is used to

determine the design and operating variables of the pipeline network for various

configurations. The applications of simulation for TPNS can be summarized as

follows:

3

1. Generate and evaluate various configurations of TPNS in order to guide in

the selection of optimal system.

2. Assist in making decisions regarding the design and operations of pipeline

network systems. During the design process, simulation could assist in

selecting the structure of the network and the geometric parameters of the

pipes which satisfy the requirements. Furthermore, it also facilitates the

selection of sites where compressors, valves, regulators and other elements

should be installed.

3. Predict the behavior of TPNS under different operating conditions.

4. Plays a vital role in analyzing the existing TPNS to study how the system

responds for future variations in demand and supply. The effect of

additional customer sites, the addition of new pipes or compressor stations

on the existing system can also be studied with the aid of simulation.

TPNS in most countries consist of a large set of highly integrated pipe networks

operating over a wide range of pressures. An increase in demand for natural gas

enhanced the development of complex TPNS. The basic problems associated with

reasonable operations of TPNS are proper supply of gas to the consumers and low

system operating costs. Proper (optimum with regard to a certain criterion)

development of transmission pipeline network system, as well as its economically

rational exploitation, are only possible if simulation procedures are applied [7].

There are three types of gas transmission networks which include gunbarrel,

branched and looped pipeline configurations [6]. The complexity of the simulation

analysis depends on the extent of the pipeline network configurations (gunbarrel,

branched, looped, etc.), the nature of the gas (single phase dry gas, two-phase gas-

liquid mixture) and other factors such as temperature of the gas, the number of

sources of the gas (single source, multi-source) and internal pipe corrosion.

4

TPNS consists of pipes and non-pipe elements such as compressors, regulators,

valves, scrubbers, etc. The simulation of TPNS system without the non-pipe

elements is relatively easier to handle and developed by Osiadacz [7]. The addition

of non-pipe elements makes the simulation of TPNS more complex due to the

modeling of the non-pipe elements. More equations have to be added into the

governing simulation equations when the non-pipe elements are considered during

analysis. Compressor station is one of the main non-pipe components of gas

transmission system and considered as a key element.

One of the basic differences among TPNS simulation analysis models with non-

pipe elements is in the way compressor station is modeled during simulation. There

have been attempts reported by various researchers on modeling compressor stations

within the TPNS during simulation [7], [16], [18], [21]. One of the options is to

consider the compressor station as a black box by setting either the suction or

discharge pressures [21]. Only little information can be obtained to be incorporated

into the simulation model to represent the compressor station. The effect of

compressor station during simulation of TPNS has been incorporated by pre-setting

the discharge pressures [7], [16], [18]. However, the speed of the compressor, suction

pressure, suction temperature, and flow through the compressor were neglected

during the analysis. Even though there have been attempts reported regarding the

simulation of TPNS with non-pipe elements, there are issues that are not addressed.

1.2 Problem Statement

Generally, from the previous approaches on TPNS simulation, it is observed that

compressor station is considered as a black box by setting few parameters or its

effect is oversimplified during simulation. This is due to the fact that the addition of

non-pipe elements makes the TPNS simulation more difficult to analyze. As a result,

5

few attempts [2], [7], [9], [16], [18] and [21]. have been done to have a complete

TPNS simulation with all its components. Since compressor station is one of the

basic elements in TPNS, the detail incorporation of all its parameters, namely: speed,

suction pressure, discharge pressure, flow rates, number of compressors and suction

temperatures is essential for a complete simulation of gas pipeline networks.

As the age of the pipe increases, the roughness of the pipe increases due to the

accumulation of various elements around the internal surface of the pipe [22]. This

might results in decrease in performance of the TPNS system, i.e. lower flow rate

capacity and higher pressure drop [23]. Limited information are available in the

literatures regarding the roughness with the age of the pipe and the performance of

the system. It is beneficial to have a TPNS simulation model to evaluate the

performance of the system with the age of the pipes.

In the petroleum industry the transportation of gas and low loads of liquids

(usually less than 0.005 holdups) occurs frequently in transmission pipelines for both

onshore and offshore operations. The liquids are usually heavy hydrocarbon fractions

and water which may be introduced from several sources. Liquids from the

compression facilities, and treatment plants as well as products of condensation may

also accompany the gas during transportation [24]. The use of single phase flow

equation for the analysis of TPNS with two-phase mixtures might lead to

underestimation of the pressure drop and flow capacity of the system. As a result, a

TPNS simulation model with appropriate flow equations which take into

consideration for the existence of liquid in the system is very useful.

6

1.3 Research Objective

The main objective of this research is to develop a TPNS simulation model for

the analysis of the performance of pipeline network system incorporating compressor

characteristics, effect of two-phase flow and the age of the pipes.

In order to achieve the objective, the thesis addresses the following issues within

the developed methodology:

1. Identifying factors that should have been considered for the design

and operation of natural gas transmission pipeline network with non-

pipe elements.

2. Formulating detailed mathematical model for the simulation which

takes into account the pipeline configurations (gunbarrel, branched

and looped), compressor characteristics, nature of the gas (single

phase, two-phase gas-liquid), temperature of the gas and the age of the

pipes.

3. Developing iterative successive substitution solution scheme for

determining the unknown pressure and flow variables and using excel

based spreadsheet to assist in evaluating different scenarios of

operation of TPNS.

4. Developing Newton-Raphson solution scheme for determining the

unknown variables and using visual C++ code to help in evaluating

the different scenarios for the design and operation of pipeline

network system.

5. Validating the Newton-Raphson simulation model with the aid of

appropriate validation techniques which includes simulations,

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comparison with previous models, and case study based on the

existing pipeline network system.

1.4 Research Scopes

The scopes of the research are summarized as follows.

1. Pipeline network system for moving gas is divided into three classes:

gathering, transmission and distribution system. Gathering pipeline

network system is responsible for collecting individual gases from gas

wells and storage system and move to the gas processing plant.

Transmission pipeline network system is used to move the gas from

gas processing plant and deliver to the distribution centers and large

industrial customers. The distribution network system is responsible

for routing of gases to individual customers. This research focuses on

the transmission system since all the gases from the gas processing

plant passes through this network system. Single source flow with one

pipe directing the gas to the compressor station is assumed as the gas

processing plant is the only source of gases to the transmission

system.

2. Development of TPNS simulation model to make performance

analysis for the gas pipeline networks involving flow capacity,

compressor ratio and power consumption for the system.

3. In a TPNS problem, the system can be modeled as steady state or

transient model depending on how the gas flow changes with respect

to time. In a steady state TPNS, the values characterizing the flow of

gas in the system are independent of time. On the other hand, transient

8

analysis requires the use of partial differential equations to describe

the relationships between parameters which make the problem more

difficult to analyze. In case of transient simulation, variables of the

system, such as pressures and flows, are function of time. This

research focuses on steady state system which is a common practice in

gas pipe line network system.

4. TPNS mainly consists of pipes and many other non-pipe devices such

as compressor stations (CS), valves and regulators. Although there are

many non-pipe elements in TPNS, CS is the key characteristics in the

network. This research focuses on addressing TPNS simulation with

CS as non-pipe element. Centrifugal compressor is assumed

throughout this study due to its frequent application in gas industry.

All the compressors within the compressor stations are arranged in

parallel.

1.5 Structure of Thesis

The remainder of the thesis is organized as follows.

Chapter 2 presents the literature review on the existing single phase optimization

and the simulation approaches for TPNS problem. It also includes the review of the

various two-phase flow models, modeling temperature variations and internal

corrosion. The basic solution methods applied in TPNS simulation are also

discussed.

The third Chapter describes the methodology used to achieve the objective

described in Chapter 1. It includes the detail description of the basic mathematical

formulation for the governing simulation equations based on pipeline configurations,

9

nature of the gas, temperature of the gas and internal corrosion of the pipes. The

detailed description of the iterative solution schemes to get the unknown variables

are also presented in this Chapter. Furthermore, the applications of the TPNS

simulation model based on successive substitution and Newton-Raphson for

modeling various network configurations are also presented.

The results of the research are discussed in detail in Chapter 4. The results of

TPNS simulation model based on successive substitution and Newton-Raphson

scheme using various TPNS configurations are presented. Moreover, the simulation

model is also tested based on the existing pipeline network system. Results of the

simulation model compared with the previous models are also discussed.

Chapter 5 presents summary of the research and the main contributions derived

from this research. Issues requiring further study are also addressed in this Chapter.

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CHAPTER 2

LITERATURE REVIEW

2.1 Introduction

For the last several decades, numerical simulations of TPNS have been carried

out at various stages of TPNS development like feasibly study, economic evaluation,

sizing, operations and monitoring of the network with various degree of success. The

simulation and optimization of TPNS have been done on the basis of different

assumptions. The process of TPNS simulation and optimization become complex

depending on the situations considered. The complexity arises from whether the

simulation or optimization model consists of single phase or multiphase flow

conditions, transient or steady state conditions, isothermal or non-isothermal

conditions and single pipe or pipeline networks conditions. Auer [25] discussed the

importance of building useful simulation model. The determining factor on how

useful the simulation model is its ability to predict the parameters of TPNS under a

wide range of conditions.

This chapter discusses the review of the most relevant literatures on the area of

single phase flow optimization by introducing the brief history of pipeline

development. The review of single phase flow simulation of pipeline networks with

non-pipe elements and with only pipe elements are also discussed. Simulation

models which consider two-phase flow analysis, temperature variations and internal

corrosion are also reviewed. The last part of this Chapter discusses the review of the

solution schemes applied for pipeline network simulations.

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2.2 Optimization of Gas Pipeline Networks

With increased applications of natural gas in power, industrial, commercial and

residential customers, the demand for natural gas is increasing for the past decades

[26]. The pipeline network gets more complex as the demand for natural gas

increases. This enhanced the importance of optimum design of pipeline network for

natural gas distribution system.

The first pipelines were built in the late 1800s to transport coal gas through cast

iron and lead pipes for street lighting. Long distance, high pressure pipeline began

operating in the United States in 1891 [27]. The development of seamless steel pipes

allowed transmission of gas at high pressure and greater quantities [28].

The world pipeline network expanded rapidly when it became apparent that

pipelines were an efficient, economic way to move oil, gas, and products to

consumers. The main contributing factors for the expansion of pipelines network

system were large discoveries of oil and gas in remote areas where there is only little

local demand. Pipelines were needed to move these supplies to market [29].

One of the main concerns with both design and operating TPNS is minimizing

the energy consumption by the system while satisfying the specified delivery

requirements throughout the system. The mathematical formulation for energy

minimization of gas transmission network is given by Mercado [6]:

Minimize ( )1.2),,(),(∑∈ snji

jiijij PPQg

Subject to

( )2.20111

=−− ∑∑∑===

w

ll

u

jj

t

ii DqQ

( )3.2222ijijji QKPP =−

[ ] ( )4.2, Ui

Lii PPP ∈

12

( ) ( )5.2,, ijjiij DPPQ ∈

where ijD represent the feasible operating domain of the compressor stations

and ),,( jiijij PPQg is the fuel consumption function at the compressor station.

In equation (2.1), the term ijQ represents the flow through the compressor station

while iP and jP are the suction and discharge pressures. Equation (2.2) is the mass

balance equation at each junction where iQ is the flow through all incoming pipes,

jq is the flow through all outgoing pipes, and lD is the load flow at each junction

node. Equation (2.3) is the pressure drop equation relating the pressures at the

upstream and downstream. The term ijK is the pipe resistance which is a function of

pipe physical properties. Equation (2.4) represents the upper and lower pressure

limits at each node while equation (2.5) is the constraints introduced for compressors

to work within the feasible domain.

From the optimization perspective, the problem of developing an optimal TPNS

is a nonlinear programming problem where the objective function, equation (2.1), is

typically nonlinear and non-convex, and some of the constraints, equation (2.3) and

equation (2.5), are also nonlinear. The problem is very difficult due to the non-

convex nature of both the objective function and the feasible region. However,

various researchers attempted different techniques in order to get the optimal

parameters of TPNS by either modifying the objective function or relaxing some of

the constraints. The following sections discuss the most commonly used techniques

in TPNS optimization.

2.2.1 Dynamic Programming

One of the techniques for TPNS optimization which has been widely applied in

previous studies since the late 1960s is dynamic programming (DP). DP approach

[30] and [31] for pipeline network optimization was first applied by Wong and

13

Larson [8]. They applied the method for fuel cost minimization of straight line

system and used recursive formulation. The gunbarrel system (Figure 2.1), which is

basically straight line type pipeline network system, possesses an appropriate serial

structure to be solved by DP.

Figure 2.1 Gunbarel structure transmission pipeline network adapted from [8]

Wong and Larson [32] then extended the principle of the application of DP from

gunbarrel type gas transmission system to a more general single source tree

structured pipeline network system. They decomposed the tree structured network

into sequential one dimensional DP problem in order to optimize the network.

For the looped structure, DP has been applied to simplified cases. Carter [3]

proposed a DP approach on more general structure with flow rates being fixed. He

developed a non sequential dynamic programming (NDP) technique which allows

pure DP to general branched and looped systems. In NDP, rather than attempting to

formulate DP as a recursive algorithm, two or three connected compressor or

regulator elements are replaced by a virtual composite element that behaves just like

its components operating in an optimal manner. Three types of composition

operations were developed in order to reduce complex system to simple one.

DP has been also combined with other techniques in order to optimize TPNS.

Ríos-Mercado et al. [6] and [33] proposed a heuristic solution procedure for fuel cost

minimization on TPNS with cyclic (looped) network configurations. The heuristic

procedure was based on a two stage iterative. At first stage, gas flow variables are

fixed and optimal pressure variables are found via DP. At a second stage, pressure

14

variables are fixed and an attempt is made to find a set of flow variables that improve

the objective by exploiting the basic network structure.

Borraz-S'anchez and R'ıos-Mercado [4] developed a hybrid meta-heuristic

solution procedure for fuel cost minimization of TPNS with cyclic topology on the

basis of DP. The heuristic methodology was based on two stage iterative procedure.

In the first stage, gas flow variables are fixed in each network arc and optimal

pressure variables in each network node were found via NDP approach developed by

Carter [3]. In the second stage, pressure variables are fixed and a short-term memory

tabu search procedure was used for guiding the search in the flow variable space.

Empirical evaluations have been made on TPNS with different configurations and

the results were compared with NDP and Generalized Reduced Gradient (GRG)

solution technique. The algorithm is capable of obtaining solution for the instances

considered. However, it is computationally expensive when compared to NDP and

GRG.

Kim [5] proposed a heuristic solution procedure for minimizing fuel cost

consumption of TPNS based on DP. The algorithm used DP as part of an iterative

procedure that modifies flow and pressure separately in an attempt to find a better

solution. The solution procedure is an iterative process. First, flow variables were

fixed by flow modification, and then DP was used to find an associated set of

pressure variables. Due to the non convexity of the objective function and the

constraints set, there is no theoretical guarantee of convergence to a local or global

minimum.

The major reason DP is attractive for optimization of TPNS problem is because it

is a general model to apply for this types of problems. Nonlinearities in the system

constraints and performance criterion can easily be handled, and constraints on both

decision and state variables introduce no difficulties. However, the application of DP

is limited to linear (gun barrel) or tree topologies. Furthermore, the computation

15

process increases exponentially with the dimension of the problem as presented in

Carter [3].

2.2.2 Mathematical Programming

Although non-linearity of the TPNS problem makes it very difficult to apply the

mathematical programming technique as a solution procedure, several researchers

have attempted to develop optimal TPNS using mathematical programming

approach. The limitation in mathematical programming is that, the nonlinear

relationships that exist either in the objective function or constraints have to be

approximated by linear relationships. This could result in a deviation from the

original problem.

Mőller [11] formulated the optimization of TPNS problem as mixed integer

linear programming. The non-linearity in both the objective function and the

constraints functions are approximated by piecewise linear functions. A separation

and branch-and-bound algorithm were developed in order to get rid off binary

variables and to fasten the calculations. The algorithm was tested on large networks

to obtain the optimal values for the TPNS. However, the non-linearity and complex

relationship between flows and pressures of pipes and compressors make the

modeling difficult for the optimization of TPNS.

Wolf and Smeers [12] proposed piecewise linear programming method to solve

the gas transmission problem. The authors formulated the problem in gas

transmission as cost minimization problem subject to the nonlinear flow pressure

relationships, material balance and pressures bounds. The solution method was based

on piecewise linear approximation of nonlinear relationships that exist during the

problem formulation. The approximated problem is solved by an extension of the

simplex method.

16

Ruz et al. [13] developed modular, constrained based model for TPNS in order to

answer questions concerning the supply, demand and transportation in the context of

optimization. The goal of the model was precise estimation of its transport capacity

to be used in a wider logistic model. The planning and scheduling for gas supply

used a mixed integer optimization model and consists of a set of interconnected

modules. Each module implements the behavior of a physical device of the gas

transmission network. The model approximates the nonlinear relationships for pipes

and compressor stations by linear functions.

Hoeven [14] formulated the gas transmission problem as constrained network

simulation model by linearizing the nonlinear relationships. The method was based

on modeling each of the elements of TPNS (compressors, pipes, valves, reducer, etc)

as constraints and relationships. The linearized equations were finally solved using

an iterative scheme.

Pratt and Wilson [10] proposed a mathematical programming approach for

optimization of the operation of TPNS. Their approach solves the nonlinear

optimization problem iteratively by linearizing the gas flow equations, constraints

and objective functions to give a linear constrained problem. The linear constrained

problem then optimized by mixed integer linear programming and solved using

branch and bound algorithm technique. The re-linearization about the optimum will

repeat until a convergence is obtained for the system. They included compressor fuel

cost and cost of flows from sources as the objective functions.

Percell and Ryan [34] proposed an algorithm using generalized reduced gradient

scheme for minimizing the fuel consumption problem for TPNS. Being a method

based on gradient search, there is no guarantee for a global optimal solution,

especially when there are discrete decision variables.

Abbaspour [9] developed a solution procedure for tackling pipeline operation

problem using simulation based optimization. The first step is to device an analysis

17

scheme that provides the simulation support required by the optimization. By gaining

some information from the simulation, the problem of gas transmission operation

was modeled as non linear programming (NLP) and solved with the sequential

unconstraint minimization technique. Since the method is based on gradient search,

there is no guarantee for global optimal solution. Furthermore, the optimization is

done at the station level rather than at network level.

Baumann et al. [15] presented a gas network optimization program (GassOpt) as

a decision support to the design of pipeline network system. The program has been

developed for transport analysis and planning purpose in order to calculate optimal

routing of natural gas in pipeline network system. The simulation program is

developed based on linear programming and optimizing engine.

Osiadacz and Gorecki [35] and Wu et al. [36] proposed solution algorithms that

tries to minimize the cost of development of pipes. The algorithms are based on

approximation of the non-linear relationships with linear constraints. A statistical

modeling technique for the design and operation of TPNS based on Monte Carlo

simulation has been proposed by Chmilar [37]. The approach is based on performing

successive approximations or estimations of the system load (flow) and capability to

generate an overview of how the system can be expected to perform.

2.2.3 Expert System

Different scholars [38]–[41] have used expert systems approach in order to

optimize the operations of TPNS. One of the main challenges that exist in applying

the expert system for optimizing TPNS operation is knowledge extraction. The

knowledge that may be extracted through interviews of experts from the natural gas

industry might not be sufficient enough to actually represent the real situations.

Sun et al. [38] proposed expert system to enhance the decision making abilities

of the dispatcher in order to optimize the natural gas operation. Sun et al. [39]

18

developed an integrated approach, both expert systems and operations research

techniques to model the operations of the gas pipelines. In the integrated model, the

expert system performs the decision to inform the dispatcher on the current system

linepack level with the associated control action to be issued and how much

horsepower should be added/reduced to satisfy future demand. On the other hand, the

decisions to turn on/off which compressor are handled by the mathematical model

adopted in the operations research. The model was tested on branched TPNS with

two compressor stations serving two customers. The optimization program

exhaustively tests all possible combinations of the compressors which increase the

complexity of the decision when the number of compressors increases.

Uraikul et al. [40] and [41] proposed an expert system as a decision support tool

to optimize natural gas pipeline operations. The expert system was developed as a

feasibility study on applicability of expert system technology to pipeline

optimizations. The three major tasks that the proposed expert system performs in

order to provide decision support to the operators are:

• Determining whether the linepack is high, low or enough,

• Suggesting the amount of break horsepower needed to increase or decrease

the linepack, and

• Recommending a compressor unit to turn on or off.

The proposed technique highly depends on the knowledge acquisition which has

been derived from historical data analysis, heuristic knowledge from expert operators

from the natural gas industry and a computer simulation model. Furthermore, the

expert system in its current version does not adequately deal with uncertain

information.

19

Chebouba et al. [42] proposed ant colony optimization algorithm to solve how to

operate TPNS efficiently. The algorithm was tested on straight line pipeline (gun

barrel) network system with five compressor stations. An optimal operating policy

for TPNS was developed for two different flow rates. The results were compared

with that of DP approach and it was reported that the results obtained were similar in

most of the cases with less computational time.

Mora and Ulieru [43] used genetic algorithm to reduce the energy used to

operate compressor stations in TPNS problem. The solution methodology was based

on two stages. The first stage of the methodology was the use of genetic algorithm

for speeding up the searching process to provide a solution in a timely manner. Each

candidate solutions generated by the search algorithm has been evaluated by

hydraulic model that simulates the steady state gas flow in the TPNS to obtain the

reaction of the system at specific control nodes and determine the feasibility of the

given solution at the second stage. Genetic algorithm has been attempted for gas

transmission network [44] and [45].

2.2.4 Analytical Hierarchical and Network Reduction

Hierarchical structure and network reduction techniques have been suggested by

various researchers as solution procedures when it is difficult to solve the gas

transmission problem in an integrated way. Analytical hierarchical approach involves

the decomposition of the gas transmission network problem into pipeline network

level and compressor station level. Some degree of success has been achieved as far

as the optimization of the compressor station subproblem is concerned. However, as

stated by Mercado [6], these approaches have limitations in globally optimizing the

minimum cost.

Osiadacz [46] developed an algorithm for optimal control of gas network based

upon hierarchical control technique. The network was divided into physically small

20

subsystems by imposing a constraint of incorporating at least one operating

compressor in each subsystem. Local problems were solved using gradient

technique. The subsystems were coordinated using “goal coordination” method to

find the overall optimum.

Wu et al. [47] developed an algorithm for optimal operation of pipeline network

system based on relaxation of the objective function (fuel cost minimization) and the

constraint (the feasible domain of the compressor station). The authors generated

four different problems for the given network and showed the differences that exist

by solving the original problem and,

• The modified problem by relaxing the feasible domain of compress stations,

• The modified problem by relaxing the fuel cost function, and

• The modified problem by relaxing both the feasible domain of the

compressor stations and the fuel cost function at the same time.

The authors considered three network problem instances. The first two problems

were a kind of network where one can get the optimal solution through exhaustive

search method. It has been reported that, lower bound solution gave an approximated

solution with good relative optimality gap for the two problem instances considered

in the paper. However, no comparison was made about the approximation of the

lower bound for the third problem as it is difficult to get the optimal solution.

Rios-Mercado et al. [48] proposed a network reduction technique for TPNS

optimization problem based on a combination of graph theory and non-linear

functional analysis. It has been reported that, the reduction technique reduced the

problem size significantly without disrupting its mathematical structure. However, no

comparison has been made to justify the applicability of the procedure and its

effectiveness compared to the existing approaches. Mohring et al. [49] presented a

21

methodology of automated model reduction for TPNS. The method is based on

computer algebra to compose automatically the model equation for different

components of TPNS (pipe, compressor, regulator, etc…). It has been reported that

the model complexity could be reduced significantly when the method is applied for

the network. However, checking of whether the parameter is within the predefined

errors is also a time consuming process.

2.3 Simulation of Gas Pipeline Networks

There are various definitions for the term simulation. One of the definitions

which is adopted in this thesis is the definition by Chung [50]. Simulation modeling

and analysis is the process of creating and experimenting with a computerized

mathematical model of a physical system. A system is defined as a collection of

interacting components that receives input and provides output for some purpose.

The simulation modeling and analysis of different types of systems are conducted

for the purposes of gaining insight into the operation of the systems. The simulation

analysis can be used for developing operating or resource policies to improve system

performance and testing new concepts. It is also used for gaining information without

disturbing the actual system and generating sample operations of the system with

different patterns and configurations [50].

Simulation of gas pipeline networks could be used to determine pressure, flow

and temperature variables of the network under different conditions. As a result,

based on the variables obtained, simulation could assist in the decisions regarding the

design and operation of the real system. Osiadacz [7] stated that at the stage of

designing TPNS, simulation could help to select the structure of the network and the

geometric parameters of the pipes which satisfy supply and demand requirements. At

22

the stage of operating TPNS, simulation could help to make various scenario

analyses in order to guide for optimal operations.

The complexity of simulation of gas pipeline network depends on the extent of

the system. Some authors performed pipeline network simulation analysis by

neglecting the non-pipe elements of the network or oversimplify their effect. The

following sections discuss the most related literatures on pipeline network simulation

with or without the non-pipe elements.

2.3.1 Simulation of Gas Networks without Non-pipe Elements

Pipeline network simulation without non-pipe elements like compressor stations,

valves, regulators is less challenging as it involves only pipes. The number of

equations and the type of equations are less compared to the simulation of pipeline

network elements with all its components. For instance, assuming constant

temperature of the gas, only pipe flow equations and mass balance equations are

involved during the simulation of the pipeline network system without non-pipe

elements. The simulation of pipeline network system without non-pipe elements are

developed by Osiadacz [7] based on graph theory.

One of the governing equations during the simulation of TPNS without non-pipe

elements is pipe flow equation. The flow of gas through pipes can be affected by

various factors such as the gas properties (specific gravity, viscosity, compressibility

and density), friction factor and the geometry of the pipes.

The relationships between the upstream pressure, downstream pressure and flow

of the gas in pipes for steady state conditions can be described by various equations

[7] and [23]. Friction factor is one of the main reasons for the variation of flow

equations.

23

In this thesis, general flow equation is adopted due to its frequent application in

gas industry [2], [5], [7] and [8] to describe the relationships between pressure

difference and gas flow in pipes. The general flow equation for the steady flow of

gas in a pipe is derived from Bernoulli’s equation. For an inclined pipe shown in

Figure 2.2, with length L and diameter D , the pressure drop between upstream node

1 and downstream node 2 of the pipe can be expressed using general flow equations

as [7]:

( )6.2264 222

522

22

1 ghTZRGPL

TPQ

DRfGZTPP

air

av

n

nn

air+⎟⎟

⎠

⎞⎜⎜⎝

⎛=−π

Hence, from equation (2.6), the flow nQ is given by

( )( )7.2

2

64

52

22

212

fGZTL

DTZR

GghPPP

PTRQ

air

av

n

nairn

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡−−

×⎟⎟⎠

⎞⎜⎜⎝

⎛=

π

If the pipe is horizontal, the elevation term )/(2 2 TZRGghP airav is zero and equation

(2.7) reduces to

( )8.2])[( 522

21

fGZTLDPP

PTCQ

n

nn

−×=

where ⎟⎟⎠

⎞⎜⎜⎝

⎛=

64

2airRC π

24

Figure 2.2 Flow of the gas though pipe adapted from [7]

Several flow equations are in use in the gas industry, all of which are

modification of the general flow equations. The differences between them depend

mainly on what expression is assumed for the friction factor f . Table 2.1 shows the

summary of the various flow equations relating the upstream node i and downstream

node j pressures for horizontal pipes. Note that for determination of the pipe flow

resistance ijK , 589.0=G , KT 0288= , kPaPn 101= , KTn0288= , KkgJRair /5.287= ,

95.0=Z are assumed throughout all equations.

25

Table 2.1 Gas flow equations and range of applications adapted from [7] and [23]

Name of Equation Range of pressure Equation

General > 700 kPa 222

ijijQKPP ji =−

51069.3 4

D

fLijK ×=

Panhandle ‘A‘ >700 kPa 854.122ijijji QKPP =−

854.4251067.1

DELKij ×=

Weymouth >700 kPa 222ijijji QKPP =−

113.7281015.8

DELKij ×=

Polyflo 75 to 700 kPa 848.122ijijQKPP ji =−

848.4251079.2

DELKij ×=

Lacey 0 to 75 kPa 2

ijijji QKPP =−

51080.1 6

D

fLKij ×=

In addition to the pipe flow equations, mass balances provide the remaining basic

equations of the governing equations for the simulation of TPNS without non-pipe

elements. The mass balance equations can be obtained based on the principle of

conservation of mass at each junction of TPNS as presented in equation (2.2).

In practical situations, the non-pipe elements like compressor stations are usually

encountered in gas transmission network systems. As a result, it is important to

address the effect of these elements into the simulation model. This study focuses on

the simulation of TPNS with non-pipe elements.

26

2.3.2 Simulation of Gas Networks with Non-pipe Elements

The addition of non-pipe elements makes the simulation of pipeline network

system more complex and difficult to handle. More equations have to be added into

the governing simulation equations when the non-pipe elements are considered

during simulation analysis. Compressor station is one of the main non-pipe

components of any gas transmission system and considered as a key elements [5] and

[6]. It adds energy to the gas in order to overcome the frictional losses and to

maintain the required delivery pressures and flows. Bloch [51] and Hanlon [52]

discussed the applications and basic principles of compressors.

One of the basic differences among TPNS simulation models is the way how

compressor station is modeled during simulation. Several attempts have been made

by various researchers on modeling compressor station within the pipeline network

system during simulation. One of the option which is suggested by Letniowski [21]

is to consider the compressor station as a black box by setting either the suction or

discharge pressures. In this case, only little information can be obtained to be

incorporated into the simulation model. Compressor station provides mass balance

equation and information regarding the suction pressure which are the two equations

added to the remaining pipe flow equations to have a complete model for the

network.

Osiadacz [7] incorporated the effect of compressor station for simulation of

pipeline network system by presetting the discharge pressures at each compressor

stations. The procedure started by making a cut-off at compressor stations. The nodes

representing compressor stations in the node data table become input nodes and the

compressor station output is denoted by an additional node. After the cut-off is made,

all nodes with preset discharge pressures become reference nodes. Reference nodes

are nodes where the pressure is known in advance of the simulation.

27

Nimmanonda et al. [16] developed a computer aided model for design of a

simulation system for TPNS which enhance iteration between the user and the

system. The simulation design program obtains the input data (pipeline pattern,

natural gas constituents, number of compressors at each station, compressors’

horsepower, pipe size, range of operating pressure, and customers’ consumption) to

generate the variables of gas properties, pressures, and flow rate of the entire pipeline

system. Continuity, mass balance, and energy balance equations were considered to

simulate the pipeline system. The simulation system can generate sample operations

of the TPNS with different configurations. However, the authors did not mention

how the compressor stations are handled during the simulation. Only TPNS without

loop was considered during the simulation model. Furthermore, the simulation model

is developed for specific geographic location which limits its applicability to the

various weather conditions.

Nimmanonda [18] developed computer aided simulation model for natural gas

pipeline network system operations where the compressor stations are modeled based

on the relationships between flow rate, break horse power (BHP), and compression

ratio. The relationship between compression ratio, BHP, and flow was developed for

three flow categories. The model needs presetting either the suction or discharge

pressure and also the speed of the compressors was not taken into considerations.

The modeling of compressor station integrated in TPNS is very difficult as the

compressor stations may contain various compressors with different arrangements.

However, there have been various attempts conducted to optimize and simulate the

operation of only the compressor stations [19], [53]-[58]. The most commonly used

compressors in gas industries are centrifugal and reciprocating. Centrifugal

compressor is assumed throughout this study due to its frequent application in gas

industry.

28

Usually, the data related to compressor are available in the form of compressor

performance characteristics as shown in Figure 2.3. In order to integrate the

characteristics of the compressor into the simulation model, it is necessary to

approximate the characteristics map with mathematical models. The basic

quantities related to a centrifugal compressor unit are inlet volume flow rate Q ,

speed n , adiabatic head H , and adiabatic efficiency η . The mathematical

approximation of the performance map of the compressor can be done based on the

normalized characteristics.

29

Inlet volume flow (m3/hr)

Isentropic Head (KJ/kg)

Figure 2.3 Typical performance map of centrifugal compressor [59]

The three normalized parameters which are necessary to describe the

performance map of the compressor includes, 2H/n , Q/n and η [60]. Based on the

normalized parameters, the characteristics of the compressor can be approximated

either by two degree [9] or three degree polynomials [2]. Three degree polynomial

which gives more accurate approximation is used in this study. Applying the

principles of polynomial curve-fitting procedures for each compressor, the

30

relationship among the basic normalized parameters can be best described by the

following two equations:

( ) ( ) ( ) ( )9.234

2321

2 nQAnQAnQAAnH +++=

( ) ( ) ( ) ( )10.234

2321 nQBnQBnQBB +++=η

where 4321 A,A,A,A and 4321 B,B,B,B are constants which depends on the unit

to define a particular compressor.

Equation (2.9) was used as basis for modeling the governing simulation equation

to represent the compressor stations. The detail modeling of compressor stations

within the TPNS will be discussed in Chapter 3.

2.4 Simulation Models with Two-phase, Temperature and Corrosion

As discussed in the previous section, one of the variations among the simulation

models is whether the model addresses the non-pipe elements or not. Apart from the

elements of the pipeline network system, TPNS simulation models have variations

among themselves based on nature of the gas. Some models assume only single

phase dry gas during the modeling of the flow equations. On the other hand, some

models perform the simulation based on two-phase gas-liquid mixture assumption.

The other variation among simulation models for gas transmission systems comes

from assumption of temperature of the gas and internal pipe corrosion during

modeling. This section presents the review of related literatures on the effect of two-

phase gas-liquid mixtures analysis, temperature, and internal corrosion during the

simulation of TPNS.

31

2.4.1 Two-phase Flow Models

Usually, in a petroleum industry both gas and low load liquids might occur in

natural gas gathering and transmission pipelines network system [61] and [62]. The

sources of the accompanying liquids could be compression facilities, treatment

plants, and/or condensation of the gas during transportation process. Several

researches have been conducted on modeling the effect of the liquids on the

transmission efficiency of the TPNS. The followings are the review of most related

literatures on the area of the transmission of gas and low load liquids.

The accompanying liquids during the transmission of gas will affect the

transportation efficiency of the system. Asante [24] suggested that, most gathering

pipelines (which typically have liquids loads up 560m3/Million m3 of gas) transport

fluids as multiphase components. On the other hand, for transmission pipelines,

where the liquid entrainment is usually less than 56m3/Million m3 of gas, most

pipeline companies typically employ “dry gas” models to predict the transport

capabilities of the system. In reality, the accompanying liquid may travel as a film or

may be distributed as dispersed droplets in the predominant gas phase. Both the film

and the droplets impede the flow of gas through the pipe. It has been reported in [62]

that, liquid loads of 5.6m3/Million m3 of gas reduced the transmission factor of the

pipeline network system by 1%.

Ellul et al. [63] presented the summary of the basic available equations in

multiphase flow conditions. The authors described the current available techniques

for the analysis of multiphase flows. The two methods for the analysis of multiphase

flow conditions are empirical approach and mechanistic approach. The former is

developed based on setting a correlation among parameters of the flow on the basis

of the experimental data for the range of conditions. The later is developed based on

the physical phenomenon of the fluids. One of the limitations that are observed in the

empirical approach is that, the method primarily aims to produce correlation valid

mainly over range of the measured data. On the other hand, the mechanistic

32

modeling approach could be used to generate relationships which are useful for wide

range of data.

Golczynski [64] emphasized on the importance of addressing the effect of

multiphase during the design of pipeline network for optimal operation of TPNS.

Asante [61] proposed the application of various two-phase flow analysis models

based on the liquid holdups. For liquid holdups less than 0.005 (typical in gas

transmission system), homogeneous approach has been recommended. When the

liquid holdups is greater than 0.005 (which is common in gas gathering pipeline

systems), stratified two-phase flow approach has been recommended.

Boriantoro and Adewumi [65] and Leksono [66] developed an integrated

single/two- phase steady state hydrodynamic model for predicting the phase change

and flow regime of fluid flow in pipes. The model predictive capability has been

tested using limited field data and published data in flow regimes. Stanley and

Vaderford [67] presented an online simulation to track the operation of two-phase

wet gas pipelines. The simulation continually receives supervisory control and data

acquisition (SCADA) updates assuring accurate conformance to the actual pipeline.

Taitel and Dukler [68] developed a mechanistic model for analytical prediction

of transition between flow regimes. The approach also provided considerable insight

into the mechanisms of the transitions. The model was tested against data mainly

collected in small diameter pipe under low pressure conditions. An attempt of flow

regime prediction for inclined, large diameter pipes has been done by Wilkens [69]

which could be used as basis for developing the pressure drop equations.

More recently, Shoham [70] developed a mathematical mechanistic model for

predicting of various two-phase flow behaviors. The model is based on the physical

phenomenon of the important flow parameters. However, the derivation of the basic

equations which represents the entire physical phenomenon requires rigorous

computations.

33

Generally, as it is reported in various literatures [24], [66], and [70], single phase

modeling approaches might not be adequate enough to predict the transport

capabilities of the pipelines when gas and liquids move as mixture. A two-phase

analysis or single phase analysis with modified friction factors may be required to

adequately predict the transport capabilities of such system.

There are various two-phase flow models reported in literatures [62], [65], and

[70]. The types of two-phase flow models that could be implemented for the system

depends on the nature and the amount of liquid that exist within the system. The

parameters of two-phase flow like the pressure drop, liquid holdup, and others are

strongly dependent on the existing flow pattern. As a result, the determination of

flow pattern is a key issue in two-phase flow analysis. The following sections discuss

the basics of flow patterns and the detail of homogeneous flow patters which is the

common flow pattern that exists in transmission pipeline network system.

2.4.1.1 Flow Pattern

The review of literatures on flow patterns reveals that there have been variations

among two-phase flow researchers on the definition and classification of flow

patterns. One of the main variation resulted from the complexity of flow phenomena

that occurred during the two-phase flow.

Flow pattern is the geometrical configuration of the gas and liquid phases in the

pipe and it occurs in a gas-liquid two-phase flow. When a gas and a liquid flow

simultaneously in a pipe, the two phases can distribute themselves in a variety of

flow configurations. The flow configurations differ from each other in the special

distribution of the interface, resulting in different flow characteristic, such as velocity

and holdup distributions. Flow pattern in a given two-phase flow system depends on

operational parameters, geometrical variables, and physical properties of the two-

phases [70].

34

Flow pattern developed by Taitel and Dukler [68] is the most commonly used

flow pattern in the analysis of two-phase flow. Figure 2.4 shows the flow patterns

existing in horizontal and near-horizontal (± 100 inclined) pipes developed by

Shoham [70].

Figure 2.4 Flow pattern in horizontal and near-horizontal pipes [70]

The existing flow patterns in horizontal and near-horizontal flow configurations

can be classified as [9] and [70]:

• Stratified smooth flow: this flow pattern occurs at relatively low gas and

liquid flow rates. The two phases separated by gravity, where the liquid-

phase flows at the bottom of the pipe and the gas-phase on the top. The

interface between them is smooth.

• Stratified wavy flow: increasing the gas velocity in a stratified flow, waves

are formed on the interface and travel in the direction of flow.

35

• Elongated bubble flow: the mechanism of the flow in the elongated bubble

flow is that of a fast moving liquid slug overriding the slow-moving liquid

film ahead of it. It occurs when the liquid slug is free of entrained bubbles

and at lower gas flow rates. This flow is sometimes referred to as plug flow.

• Slug flow: occurs at higher gas flow rates, where the flow at the front of the

slug is in the form of an eddy which entrained bubbles.

• Annular flow: this flow occurs at a very high gas flow rate. The gas-phase

flows in a core of high velocity, which may contain entrained liquid droplets

and the liquid flows as a thin film around the pipe wall.

• Wavy annular flow: this flow occurs at the lowest gas flow rates. Most of

the liquid flows at the bottom of the pipe while aerated unstable waves are

swept around the pipe periphery and wet the upper pipe wall occasionally.

This flow occurs on the transition boundary between stratified way, slug and

annular flow.

• Dispersed bubble flow: this flow is one of the homogeneous flows where

either of the two-phases flowing simultaneously in the pipeline is completely

dispersed in the other. Dispersed bubble flow occurs at very high liquid rates.

On the other hand, at a very high gas rates coupled with low liquid loading

the flow is termed as mist flow. In both dispersed bubble flow and mist flow,

the two phases move at the same velocity, and the flow is considered

homogeneous with no-slip.

2.4.1.2 Homogeneous Flow Model

The homogeneous (dispersed bubble and mist) flow models treat the gas-liquid

mixtures as a pseudo single phase with average fluid properties [70] and [71]. It was

reported in [61], low load liquids ( usually less than 0.005) holdup exist in natural

36

gas transmission pipeline network systems. Hence, the best flow pattern to describe

the two-phase gas-liquid mixtures in gas TPNS is mist flow. Because of this,

homogeneous (pseudo single phase) approach is applied in this thesis to describe the

behavior and transport properties of the system of gas and low loads of liquids in the

pipeline.

The model for homogenous two-phase flow analysis is developed based on the

assumption of compressible flow, variable cross-sectional area of pipe, variable

quality of the gas along the length of the pipe [70]. The conservation equations of

mass, momentum, and energy for the model are developed using a control volume

with a cross-sectional are PA and differential length dL in the axial direction as

shown in Figure 2.5.

Figure 2.5 Schematic of homogeneous no-slip flow model adapted from [70]

The continuity equation for the mixture is given by

( )11.2ConstantAvM Pmm == ρ

where M is the total mass flow rate, and mρ and mv are the mixture average

density and velocity, respectively.

A momentum balance on the control volume can be defined as

37

( )12.2sinθρτ gASdLdpA

dLdvM mPwPP

m −−−=

where L is the axial direction, P is the pressure, wτ is the shear stress at the

wall, PS is the pipe perimeter, and θ is the inclination from the horizontal.

Dividing equation (2.12) by the cross-sectional area PA and solving for the

pressure gradient yields

( )13.2sindLA

dvMgAS

dLdp

P

mmw

P

P ++=− θρτ

As shown in equation (2.13), the total pressure gradient equation is composed of

three components: frictional, gravitational, and acceleration components. This

equation will be further developed in Chapter 3 to enable the calculation of each of

the pressure gradient components and the total pressure gradient which provide one

of the governing simulation equations.

2.4.2 Simulation Model with Temperature Variations

One of the variations among the different models developed for the analysis of

TPNS is whether the model takes into account the effect of temperature or not. The

model for the analysis of TPNS can be done on the basis of constant gas flow

temperature i.e. isothermal condition. In reality, as indicated in Menon [23], the

temperature of the gas in TPNS varies along the length of the pipeline due to heat

transfer between the gas and the surrounding soil.

Osiadacz and chaczykowski [72] made a comparison of flow of gas in gunbarrel

pipelines under isothermal and non-isothermal conditions. A significant pressure

profile difference along the pipeline was reported between the isothermal and non-

isothermal analysis. This difference increases with increase in quantity of gas

transmitted. Thermal model for single pipe is also presented in [73].

38

For the variation of gas temperature in TPNS, Menon recommended the

calculation of pressure drop to be done by considering short lengths of pipe that

make up the total pipeline. Figure 2.6 shows a buried pipeline transmitting gas from

node A to node B. Considering a short segment with length of LΔ from the pipeline,

the variation of temperature along the pipeline can be analyzed by applying the

principles of heat transfer.

Figure 2.6 Analysis of temperature variations adapted from [23]

When the gas flows from upstream end with temperature of 1T to downstream

end of the pipe segment with temperature of 2T , the temperature of the gas will drop.

On the other hand, there will be heat transfer between the gas and surrounding due to

temperature difference. The temperature equation relating the two nodes 1 and 2 of

the segment can be expressed as [23]

( )14.2)( 12θ−−+= eTTTT ss

( )15.2PMCLUDΔ

=πθ

where sT is the average soil temperature surrounding pipe segment, M is

mass flow rate of the gas, PC is average specific heat of gas, U is the overall

heat transfer coefficient, and D is the diameter of the pipe.

It can be seen from equation (2.14) that as the pipe length increases, the term θ−e

approaches zero and the temperature 2T becomes equal to soil temperature, sT .

39

Therefore, in a long gas pipeline, the gas temperature ultimately equals the

surrounding soil temperature.

The temperature of the gas is also affected by the compression process. In

adiabatic compression of natural gas, the final temperature of the gas can be

determined knowing the initial temperature and initial and final pressures. From the

adiabatic compression equation and perfect gas law, the discharge temperature is

given by [23, 60]:

( )16.2

1

1

2

2

1

1

2 kk

PP

ZZ

TT

−

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛

where 1Z and 2Z are gas compressibility factor at suction and discharge side

respectively and k is the specific heat ratio.

From equations (2.14) and (2.16), it can be seen that the variations of

temperature in gas TPNS are controlled by these equations. Hence, these equations

will be further developed in Chapter 3 to be incorporated into the governing

simulation equations.

2.4.3 Internal Corrosion

Corrosion in oil and gas industries is one of the serious challenges which affect

the performance of the pipeline network system. As presented in [74], corrosion is

the destructive attack of a metal by chemical or electrochemical reaction with its

environment. The phenomenon of corrosion involves reactions which lead to the

creation of ionic species, by either loss or gain of electrons. Corrosion of material

depends on several factors. Some of them includes: nature of the material or alloy,

surface condition/roughness, composition, moisture absorptivity, environment,

temperature, humidity and corrosive elements.

40

It has been reported in [22] that an increase in ages of the pipes resulted in higher

pipe roughness due to the accumulation of various elements around the internal

surface of the pipe. Figure 2.7 shows the effect of service life of the pipe on

roughness. An increase in roughness has major effect on the performance of the

transmission system i.e. lower flow rate capacity and high pressure drop as indicated

in [23].

Figure 2.7 Effect of years in service on pipe roughness [22]

So far, limited information is available from literature about how the roughness

of the pipes varies with the age of the pipe. However, there are several studies

conducted on corrosion of pipelines for both single phase flow and two-phase gas-

liquid mixtures flow [75]-[78]. One of the issues addressed in this thesis is that, the

effect of ages of the pipes on the performance of TPNS has been incorporated within

the simulation model.

41

2.5 Solution Principles Applied in Gas Pipeline Network Simulations

The selection of an appropriate solution scheme is essential in order to get the

required parameters which are necessary to analyze the performance of the TPNS.

The most commonly used solution schemes for analyzing the governing equations of

the simulation are successive substitution (SS) and Newton-Raphson (NR) solution

schemes[79].

Successive substitution scheme [79] and [80] has been applied for simulation of

water pumping system [81] and for obtaining solutions for boundary value problems

[82]. The major advantage of using successive substitution scheme is that the method

is easy to use when the numbers of equations within the system are few.

Newton-Raphson scheme [79]-[81] has been also applied in gas pipeline network

simulation and other networks for various analysis [83]-[86]. One of the major

advantages of Newton-Raphson algorithm is that the method is more reliable and

rapidly convergent. Furthermore, it is not necessary to list the equations in any

special order. Abbaspour et al [83] used the Newton-Raphson algorithm to solve the

nonlinear finite difference thermo-fluid equations for two-phase flow in a pipe. Brkic

[84] used Newton-Raphson method to solve the nonlinear flow equations in natural

gas distribution networks. Beck and Boucher [85] also applied the Newton-Raphson

algorithm to analyze steady state fluid circuits. Kessal [86] used the Newton-

Raphson algorithm for analyzing fluid flow in gas pipelines.

The maximum relative percentage error at each iteration for both successive

substitution and Newton-Raphson is calculated based on the relationships given as

[79]:

( )17.2100)]max(/)([max% ×−= newoldnew XXXErrorageMaximum where X is the vector that represents the unknown variables.

42

In order to take the advantages of both solution schemes, two solutions are

developed in this thesis. The first one is based on successive substitutions and

applied for simple TPNS configurations. The second one is a generalized solution

scheme based on Newton-Raphson which is applied for complex TPNS

configurations. Detailed analysis of the solution schemes for the application of TPNS

simulation is discussed in Chapter 3.

2.6 Summary

This Chapter presented the review of optimization of gas pipeline network

systems, simulation of gas pipeline networks, the simulation models with two-phase

flow, temperature variations and corrosion, and solution principles applied in gas

pipeline network simulation.

The various TPNS simulation models revised in this thesis are summarized in

Table 2.2. From the review of the TPNS simulation models, it is observed that while

there is significant progress on the area of TPNS simulation methodologies, there are

still issues that need to be addressed. The effects of the compressor characteristics,

two-phase flow and internal corrosion during TPNS simulation modeling are

addressed in this research.

43

Table 2.2 Summary of TPNS Simulation Models

Method Author Main features Limitations

Newton-loop Osiadacz [7] • Preset the discharge pressures at each compressor stations.

• Single phase flow and constant temperature

• Speed, discharge pressures, suction temperature neglected

• Constant corrosion Black box Letniowski [21] • Consider the compressor station as a black

box • Single phase flow, constant temperature

• Speed, discharge pressures, suction temperature neglected

• Constant corrosion Graph theory Wu et al. [47] • Describing the feasible region of

compressors mathematically and optimizing the operation of TPNS.

• Single phase flow , constant temperature

• Based on approximation of fuel cost functions.

• Constant corrosion

Correlation Nimmanonda

et al [16] and [18]

• Continuity, mass balance, and energy balance equations used.

• The relationship between compression ratio, BHP, and flow was developed for three flow categories.

• Single phase flow, constant temperature

• Compressor stations handling and only TPNS without loop

• Requires presetting either the suction or discharge pressure

• Speed of the compressors was not taken into considerations.

Mathematical approximation

Abbaspour [9] • Use of polynomial approximation for modeling the compressor to develop simulation based optimization.

• Two phase flow and variable temperature

• Perform optimization or simulation on station level.

• Constant corrosion

44

Most of the studies on optimization of pipeline network system used either

relaxation of the constraints or approximation of the constraints with equivalent

constraints to facilitate for computation. These could result in deviation from the

original problem. Furthermore, the optimization of pipeline gives only the optimal

operation of the system. On the other hand, simulation of the pipeline network

system is used to investigate the off design operation of the system.

The review of simulation models developed for natural gas TPNS indicated that

the models were either limited for pipe only or treat the non-pipe elements based on

simplified approach by neglecting the importance of the parameters such as speed,

suction pressure, discharge pressure, and suction temperatures. In some cases, the

applications of the models may be limited to either simple pipeline configurations or

fixed environmental conditions. Pressure drop and flow rate of the gas may be

affected as the age of the pipe increase. However, limited studies on the relationships

between the age of the pipe and its effect on pressure drop and flow are reported on

literatures. The study conducted on non-isothermal model of TPNS lies mostly on

single pipe or limited to simplified network configurations like gunbarrel.

In the following Chapter, the details of the development of the methodology to

address the objective of this research are presented. The basic principles adopted in

this Chapter are utilized to develop the methodology.

45

CHAPTER 3

METHODOLOGY

3.1 Introduction

In this Chapter, a suitable methodology was developed to achieve a TPNS

simulation model for performance analysis of gas pipeline networks. The

determination of operational and design variables for the TPNS were conducted

based on the analysis of factors involved during the transmission process.

Appropriate mathematical formulations and suitable solution procedures were

established to get reliable simulation results that predict the actual situations. The

basic equations for the TPNS simulation were derived from the principles of flow of

fluid through pipe, compressor characteristics and the principles of mass balance at

the junction of the network.

First, the mathematical formulation of the TPNS simulation model is discussed.

The basic single phase flow equations, compressor characteristics equations, mass

balance and looping conditions which form the governing simulation equations are

presented. Then, the solution schemes to get the flow and pressure variables based on

iterative successive substitution and Newton-Raphson schemes are presented. The

enhanced Newton-Raphson based TPNS simulation model which contains additional

features such as two-phase flow analysis, temperature variations and internal

corrosion is also discussed. The last two sections present the performance evaluation

of TPNS and summary of the Chapter. Figure 3.1 shows the structure of the

simulation model for natural gas TPNS.

46

Figure 3.1 Structure of the simulation model for transmission pipeline networks

47

3.2 Mathematical Formulation of the Simulation Model

The mathematical formulation and the types of equations incorporated into the

governing simulation equations depend on the configurations of the network and

elements of the TPNS as shown in Figure 3.1. The mathematical model for the TPNS

simulation is developed based on equations which govern the flow of the gas through

pipes, the performance characteristics of the compressors, and the principles of

conservation of mass.

3.2.1 Single Phase Flow Equations Modeling

One of the governing equations for the simulation is derived based on the

principle of flow analysis of gas in pipes. The flow of gas through pipes can be

affected by various factors such as the gas properties, friction factor, and the

geometry of the pipes. As discussed in section 2.3.1, several flow equations are in

use in the gas industry for single phase gas flow. In general, for a horizontal pipe

connecting node i and j (Figure 3.2), the general flow equation for relating upstream

pressure iP , downstream pressure jP and the flow through pipe ijQ can be expressed

as:

( )1.3222ijijji QKPP =−

where ijK is to be determined by the gas properties and the characteristics of

pipe connecting node i and j .

48

Figure 3.2 Pipe joining two consecutive nodes

Equation (3.1) can be represented as functional form [81]. If all nodal pressures

and flow rate are unknown, the functional representation of equation (3.1) takes the

form:

( )2.30),,( =ijji QPPf

If the upstream pressure and the flow rate are the only unknowns, the functional

representation for equation (3.1) takes the form

( )3.30),( =iji QPf

Note that functional form of representation for an equation consists of only

parameters which are unknown in that equation. Figure 3.3 shows part of the TPNS

consisting of three compressor stations (CS1, CS2 and CS3) and six pipes i.e. pipe 0-

1, 2-3, 3-4, 5-6, 3-7 and 8-9.

49

Figure 3.3 Part of transmission pipeline network system

Assuming that the source pressure at node 0 and the demand pressures at node 6

and 9 are known, the general flow equations for the pipes for single phase flow and

the corresponding functional representation can be summarized as shown in

Table 3.1.

Table 3.1 Single phase flow equations and functional representations

Pipe node General flow equation Functional representation

Start node End node

0 1 2101

21

20 QKPP =− 0),( 111 =QPf

2 3 2123

23

22 QKPP =− 0),,( 1322 =QPPf

3 4 2234

24

23 QKPP =− 0),,( 2433 =QPPf

3 7 2337

27

23 QKPP =− 0),,( 3734 =QPPf

5 6 2256

26

25 QKPP =− 0),( 255 =QPf

8 9 2389

29

28 QKPP =− 0),( 386 =QPf

For ease of representation, from now onwards, the term flow equation will be

frequently used to mean the general flow equation. Furthermore, the functional form

of representation will be used interchangeably with the flow equation.

50

3.2.2 The Looping Conditions

When the TPNS contains loops, additional equations must be incorporated to the

flow equations for the pipes. These additional equations are obtained from looping

condition. The looping condition states that for each closed loop within the network

system the pressure drop is zero [7, 23].

Looped piping system, as shown in Figure 3.4, consists of two or more pipes

connected in such a way that the gas flow splits among the branch pipes and

eventually combine downstream into a single pipe. It can be constructed of the same

diameter pipe as the main pipeline or based on different size. The reason for

installing loops is to reduce pressure drop in a certain section of the pipeline due to

pressure limitation or for increasing the flow rate in bottleneck sections. For instance

for the TPNS shown in Figure 3.4, by installing a pipe loop from node 3 to node 6,

the overall pressure drop can be reduced due to the split of flow rate through the two

pipes. Based on the looping condition, the pressure drop in pipe branch 3-C1-4 must

equal the pressure drop in pipe branch 3-C2-4. This is due to the fact that both pipe

branches have a common starting point (node 3) and common ending point (node 4).

51

Figure 3.4 Looped pipeline network system

The pressure drop due to friction for single phase flow in branch 3-C1-4 can be

formulated based on the general flow equation as

( )4.351

22112

62

3 DQLKPP =−

where, 1K is a parameter that depends on gas properties and friction factor,

2Q is the flow rate, 1L and 1D are the length and diameter of the pipe branch

3-C1-4, respectively.

Similarly, the pressure drop due to friction for single phase flow in branch 3-C2-4

can be given as

( )5.352

23222

62

3 DQLKPP =−

where, 2K is a parameter that depends on gas properties and friction factor,

3Q is the flow rate, 2L and 2D are the length and diameter of the pipe

branch 3-C2-4, respectively.

52

In equations (3.4) and (3.5), the constants 1K and 2K are equal, since the same

gas is flowing through both branch pipes. Combining both equation results in

( )6.352

232

51

221

DQL

DQL

=

Further simplification of equation (3.6) gives

( )7.35.2

2

15.0

1

2

3

2⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

DD

LL

QQ

Equation (3.7) is referred to as the looping condition and it is one of the

governing simulation equations when the pipeline network system contains loops.

For the unknown flow rate 2Q and 3Q , the functional representation for equation

(3.7) takes the form

( )8.30),( 32 =QQf

3.2.3 Compressor Characteristics Equations

When the TPNS contains non-pipe elements, one of the governing simulation

equations should have to be derived from compressor. Compressors are usually

characterized by the performance map. In order to integrate the performance

characteristics of compressor to the governing simulation equations, each of the

constant speed curves from the map should have to be represented by mathematical

model. Even though this type of representation results an accurate approximation for

the curves, it is time consuming. Hence, the compressor map is usually represented

based on single curve by using normalized parameters.

53

Figure 3.5 shows the plot of normalized head against the normalized flow rate for

typical centrifugal compressor data taken from [55] for speeds of 8856, 8000, 7000,

6000, 5000, 4000, and 3630rpm. The corresponding plot for efficiency against the

normalized flow rate is shown in Figure 3.6. It can be seen from both the figures that

all the constant speed curves had the tendency to coincide as single line.

Figure 3.5 Normalized head and normalized flow rate for different speeds of a

typical centrifugal compressor

54

Figure 3.6 Efficiency and normalized flow rate for different speeds of a typical

centrifugal compressor

In considering the effect of compressors for the TPNS simulation model, the

relationships as in equation (2.9) and equation (2.10) might not be used directly. The

information from the compressor map should have to relate the discharge pressure,

the suction pressure and flow rate. The relationships between suction pressure sP ,

and discharge pressure dP with the head H is given as [60] :

( )9.31⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−⎥⎦

⎤⎢⎣

⎡=

m

s

dS

PP

mZRT

H

where ( ) kkm /1−= with k to be specific heat ratio, R is gas constant, ST is

the suction temperature and Z is the compressibility of the gas.

55

Substituting the value of H from equation (3.9) into equation (2.9) and

rearranging yields the required compressor performance equation which can be

incorporated as governing equation for the simulation model.

( ) ( ) ( ){ } ( )10.31/// 34

2321

2++++=⎟⎟

⎠

⎞⎜⎜⎝

⎛nQAnQAnQAA

ZRTmn

PP

S

m

s

d

Equation (3.10) represents a general compressor equation for single compressor.

It can be seen that most of the parameters that describe the compressor are

incorporated in the general compressor equation. This is one of the significant

contributions in the area of simulation of TPNS with compressor stations as non-pipe

elements.

If the suction side pressure, discharge side pressures and flow rates are unknown,

equation (3.10) can also be represented with short functional form as

( ) ( )11.30,, =QPPf sd

The coefficients 4321 A,A,A,A used in equation (2.9) and the corresponding

coefficients 4321 B,B,B,B for equation (2.10) are unique and depend on the units used

in the compressor map. For gas pipeline compressors taken from [55] and [59], the

different coefficients for equation (2.9) and (2.10) are summarized as shown in

Table 3.2.

56

Table 3.2 Values of coefficients for compressor equations

Types of compressor Coefficients for

equation (2.9)

Coefficients for

equation (2.10)

Centrifugal compressor

from [55]

A1 1.20 x 10-06 B1 9.7 x 10-01

A2 -2.48 x 10-09 B2 -1.14 x 10-02

A3 -4.60 x 10-12 B3 2.56 x 10-04

A4 -3.17 x 10-13 B4 -1.51 x 10-06

Centrifugal compressor

from [59]

A1 1.08 x 10-06 B1 7.99 x 10-01

A2 -1.96 x 10-08 B2 -1.03 x 10-02

A3 4.71 x 10-10 B3 6.19 x 10-04

A4 -5.83 x 10-12 B4 -7.86 x 10-06

Equation (3.10) represents a general compressor equation for single compressor.

However, there are cases where several compressors may work within compressor

stations either on the basis of serial or parallel arrangements. When the compressors

operate in series within the station as shown in Figure 3.7, compressor equations for

each compressor should have to be developed to represent the effect of all

compressors.

Figure 3.7 Compressors operating in series within a station

For instance, if there are cn number of compressors operating in series within the

station, there will be cn number of independent compressor equations for the station.

57

Equation (3.12) shows the compressor equation for the first compressor within the

station.

( ) ( ) ( ){ } ( )12.313/42/3/21

2

1

1 ++++=⎥⎦

⎤⎢⎣

⎡nQAnQAnQAA

sZRTnm

PP

m

s

d

where 1sP and 1dP are the suction and discharge pressures for the first

compressor.

The remaining compressor equations within the station were developed by

following the same procedure as in equation (3.12).

For compressors operating in parallel within the stations as shown in Figure 3.8,

only a single equation may represent the compressor station assuming that identical

compressors are working within the station. For instance, for the compressor station

with cn number of compressors working in parallel, the general compressor equation

can be modified to represent the compressor station as

[ ] [ ] [ ]{ } ( )13.313)/(42)/(3)/(21

2+++=⎥

⎦

⎤⎢⎣

⎡ncnQAncnQAncnQAA

ZRTnm

PP

S

m

s

d

where sP and dP are the suction and discharge pressures for the compressor

station.

58

Figure 3.8 Compressors working in parallel within a station

The general compressor equation was validated based on the performance map

of the compressor. Figure 3.9 shows a comparison of the plot of the performance

characteristics of the compressor generated by Equation 3.10 and actual data

collected from the performance map of the compressor shown in Figure 2.3. It is

observed that the maximum percentage error introduced when the performance of

the map is approximated by the general compressor equation was 3%.

59

Figure 3.9 Comparison of selected data and approximated data based on the

compressor data from [59]

Figure 3.10 shows the plot of the approximated pressure ratio ( )PsPd / based on

Equation 3.10 against actual pressure ratio from the characteristic curve for the

typical compressor taken from [55] at speed of 8000 rpm. As the speed of the

compressor increased, the deviation between the actual and calculated pressure ratio

increased which made the approximation a bit far from the real values. For gas

pipeline centrifugal compressor where the maximum pressure ratio is limited to

around 1.5, the mathematical model developed in Equation 3.10 could be taken as

reasonable approximation for the performance characteristics.

60

Figure 3.10 Comparison of calculated and actual pressure ratios

3.2.4 Mass Balance Equations

In addition to the pipe flow equations, looping conditions, and compressor

equations, mass balances provide the remaining basic equations for the simulation of

TPNS configurations with branches. The mass balance equations were obtained

based the principle of conservation of mass at each junction of TPNS.

At any junction node c within the TPNS, shown in Figure 3.11, the generalized

mass balance equation for t incoming pipes, u outgoing pipes, and w load pipes can

be summarized as:

( )14.3011 1

=−− ∑∑ ∑=

=

=

=

=

=

wl

ll

ti

i

uj

jji DqQ

where tQQQ ,....,, 21 are flow through incoming pipes to junction c , uqqq ,....,, 21 are flow through outgoing pipes from junction c and lDDD ,....,, 21 are the

load from junction node c .

61

Figure 3.11 Mass balance formulation

The functional representation of equation (3.14) takes the form shown in

equation (3.15) if all the flow rates through the incoming and outgoing pipes are

unknown.

( )15.30)...,,,,...,,,( 2121 =ut qqqQQQf

3.3 Successive Substitution based TPNS Simulation Model

In section 3.2, the details of the mathematical formulation for a given TPNS have

been presented. The number of variables (flow and pressure) to be determined

depends on the configurations (number of pipes, number of compressor stations,

number of branches, and number of loops). The principles of flow through pipes,

performance characteristics of the compressor, conservation of mass, and looping

conditions were able to provide sufficient number of equations to determine the

unknown variables.

Once the basic governing simulation equations for TPNS are developed, the

result for unknown pressure and flow variables could be determined. The most

commonly used solution schemes for analyzing the governing equations of the

simulation are iterative successive substitution and Newton-Raphson solution

schemes. The major advantage of using successive substitution scheme is that the

method is easy to use when the numbers of equations within the system are few. One

62

of the advantages of Newton-Raphson algorithm is that the method is more reliable

and rapidly convergent. Furthermore, it is not necessary to list the equations in any

special order.

In order to take the advantages of both solution schemes, two solutions are

developed in this thesis. The first one is based on successive substitutions and

applied for simple TPNS configurations. The second one is a generalized solution

scheme based on Newton-Raphson which is applied for complex TPNS

configurations.

A relationships between the number of pipes np, number of compressor stations

ns with compressors working in parallel, number of loops nl, and the number of

branches (junctions) nj with the total number of unknown pressure and flow

variables are developed. Table 3.3 shows the summary of the relationships between

the number of equations available and the total number of unknown variables.

Table 3.3 Number of equations and unknowns for TPNS configurations

Items No. of equations No. of unknowns

Flow

Comp. Mass balance

Looping conditions

P

Q

Basic configurations and elements of TPNS

Pipes np - - - (np+ns) – (nj+1)

2nl + 2nj + 1

Comp. - ns - - Branches - - nj - Loops - - - 2nl

Total number of equations and unknowns

np + ns + nj +2nl

np + ns + nj +2nl

The data shown in Table 3.4 were used throughout the analysis for the numerical

evaluations. All the properties of the gas were collected from the nearest operational

gas transmission company. For the compressor station, all units within the stations

are assumed to be identical and are connected in parallel. The data related to pipe

diameters, lengths and customer requirements are based on existing TPNS and

literatures. The dimensions used for the parameters are ][kPaP , ][KT , ][kmL , ]/[ 3 hrmQ

and ][mmD . The pipe flow resistance 50960.7 DfLEKij ×+= .

63

Table 3.4 Gas properties

In this section, successive substitution based TPNS simulation model is

discussed. The application of the model is demonstrated based on two pipeline

network configurations.

When the successive substitution scheme is to be applied for the determination of

unknown variables for the given TPNS, information flow diagram (IFD) is an

important tool [81]. IFD involves the representation of the basic equations of the

TPNS as an input-output information blocks and arranging them in such a way that

only one output can be calculated from each block. A compressor might appear in

TPNS diagram like the one shown in Figure 3.3. But, the information flow block of

the compressor might take one of the forms shown in Figure 3.12. Note that the

blocks in this figure represent a compressor equation presented in function form as

( ) 0,, 21 =PPQf . In order to develop the IFD for the TPNS, all the components of the

network should have to be represented as information block and arranged in such a

way that only one output can be determined from single block.

Gas properties Numerical values used for analysis

Gas composition

Methane 92%

Ethane 5%

Nitrogen 1%

Others 2%

Gas gravity (G) 0.5

Gas flowing temperature (T) 308 K

Base pressure (Pn) 101kPa

Base temperature (Tn) 288K

Gas constant for air (Rair) 287.5J/kgK

Gas compressibility (Z) 0.91

Isentropic exponents (k) 1.287

64

Figure 3.12 Possible information flow blocks for compressor

The method of successive substitution is closely associated with the IFD of the

system. The solution procedure starts by assuming a value of one or more variables,

beginning the calculation, and proceeding through the system until the originally

assumed variables have been recalculated. The recalculated value is then substituted

successively and the calculation loop is repeated until satisfactory convergence is

achieved. Figure 3.13 shows the developed flowchart of the simulation model based

on the iterative successive substitution scheme.

65

Figure 3.13 The developed flow chart for successive substitution based TPNS

simulation model

66

3.3.1 Case 1A: Single Compressor Station and Two Customers Module

In single compressor station two customers network, it is required to transmit gas

from source to two different customers with the required pressure and flow rate.

Figure 3.14 shows schematic diagram of the TPNS when the gas is delivered from

source to two different customer sites of station D1 and D2 using single compressor

station (CS).

Figure 3.14 Gas transmission system for two customers

The TPNS consists of 4 pipes, 1 compressor station, 1 junction, and with no loop.

Therefore, np = 4, ns = 1, nl = 0, and nj = 1. As a result, based on Table 3.3, there are

3 nodal pressures and 3 flow variables to be determined. A total of 6 independent

equations were obtained in order to solve the network problem. The basic governing

equations for TPNS were developed from pipe flow equations, compressor stations

equations and mass balance equations based on the discussion in section 3.2. There

are 4 pipe flow equations, 1 compressor station equation and 1 mass balance

equation which form the required number of equations to solve for the unknown

variables.

The summary of flow equations and their corresponding functional

representation for the given TPNS are shown in Table 3.5.

67

Table 3.5 Summary of flow equations and their corresponding functional

representations for the given TPNS

Pipe node Flow equation Functional representation

Start node End node

0 1 2101

21

20 QKPP =− 0),( 111 =QPf

2 3 2123

23

22 QKPP =− 0),,( 1322 =QPPf

3 D1 2113

21

23 CDD QKPP =− 0),( 133 =CQPf

3 D2 2223

22

23 CDD QKPP =− 0),( 234 =CQPf

The compressor equation for the TPNS is given as:

( ) ( ) ( ) ( )16.31]///[ 314

213121

2

1

2 ++++=⎟⎟⎠

⎞⎜⎜⎝

⎛nQAnQAnQAA

ZRTmn

PP

S

m

The corresponding functional representation for equation (3.16) takes the form

( ) ( )17.30,, 1215 =QPPf

The mass balance equation is given by:

( )18.3211 CC QQQ +=


( ) ( )19.30,, 2116 =CC QQQf

Based on the functional representations for the mathematical formulation of the

governing simulation equations for the given TPNS, the IFD can be drawn as shown

in Figure 3.15. Using this IFD, the successive substitution solution scheme could be

started by assuming one or more variables. For the given TPNS, the solutions can be

obtained by assuming initial value for 1Q and 3P . Once these variables are assumed,

the value of 2CQ can be calculated, then 1CQ , 2P , etc. by following the IFD. The

68

iteration will continue till the desired error limits or the number of iterations is

achieved.

Figure 3.15 IFD of pipeline network system with two customers

The application of the SS based TPNS simulation model for the given network

was conducted based on nodal pressure requirements and pipe data shown in Table

3.6 and Table 3.7 , respectively.

Table 3.6 Pressure data for single CS and two customers

Node Pressure[kPa]

D1 1800

D2 1800

Source 4000

Table 3.7 Pipe data for single CS and two customers

Pipe node Diameter [mm] Length [km]

Start node End node

0 1 400 80

2 3 400 60

3 D1 400 80

3 D2 300 60

69

3.3.2 Case 2A: Two Compressor Stations and Two Customers Module

The problem in two compressor stations and two customer network is to transmit

gas from source to two customers using two compressor stations. Figure 3.16 shows

TPNS when gas is delivered from source to two different customers’ sites designated

as 1D and 2D using two compressor stations.

Figure 3.16 Pipeline network with two CSs and two customers

This TPNS consists of 5 pipes, 2 compressor stations, 1 junction, and with no

loop. Therefore, np = 5, ns = 2, nl = 0, and nj = 1. As a result, based on Table 3.3,

there are 5 nodal pressures and 3 flow parameters to be determined. Therefore a total

of 8 independent equations should have to be obtained in order to solve the network

problem. The basic governing equations for TPNS were developed from pipe flow

equations, compressor stations equations and mass balance equations based on the

discussion in section 3.2. There are 5 pipe flow equations, 2 compressor station

equations and 1 mass balance equation which formed the required number of

equations to solve for the unknown parameters.

The summary of flow equations and their corresponding functional

representation for the given TPNS are shown in Table 3.8.

70

Table 3.8. Summary of flow equations and their corresponding functional

representations for the given TPNS


Start node End node

0 1 2101

21

20 QKPP =− 0),( 111 =QPf

2 3 2123

23

22 QKPP =− 0),,( 1322 =QPPf

4 5 2145

25

24 QKPP =− 0),,( 1543 =QPPf

5 D1 2115

21

25 CDD QKPP =− 0),( 154 =CQPf

5 D2 2225

22

25 CDD QKPP =− 0),( 255 =CQPf

The compressor equations for the given TPNS were summarized in equation and

function forms as shown in Table 3.9.

Table 3.9 Summary of compressor equations and functional representations

Compressor

stations

Compressor equation Functional representation

s

CS1 ( ) ( ) ( ) 1]///[ 3114

21131121

21

1

2 ++++=⎟⎟⎠

⎞⎜⎜⎝

⎛nQAnQAnQAA

ZRTmn

PP

S

m

0),,( 1216 =QPPf

CS2 ( ) ( ) ( ) 1]///[ 3214

22132121

22

3

4 ++++=⎟⎟⎠

⎞⎜⎜⎝

⎛nQAnQAnQAA

ZRTmn

PP

S

m

0),,( 1437 =QPPf

Mass balance equation at junction node 5 is given by:

( )20.3211 CC QQQ +=


( ) ( )21.30,, 2118 =CC QQQf

71

Based on the functional representations for the mathematical formulation of the

governing simulation equations for the given TPNS, the IFD is as shown in

Figure 3.17. Similar procedure was followed as in the case of single compressor

station with two customers. Using this IFD, the successive substitution solution

scheme can be started by assuming two variables. For the given TPNS, the solution

is obtained by assuming initial value for 1Q and 5P . Based on the assumed variables,

the value of 2CQ will be calculated, then 1CQ , 4P , etc. by following the IFD. The

iterations will be continued till the desired error limits or the number of iteration is

achieved.

Figure 3.17 IFD for pipeline network with two CSs

The application of the SS based TPNS simulation model for the given network

was conducted based on nodal pressure requirements and pipe data shown in Table

3.6 and Table 3.10 , respectively.

72

Table 3.10 Pipe data for two CSs and two customers

Pipe node Diameter [mm]

Length [km] Start node End node

0 1 200 100 2 3 200 50 4 5 200 100 5 D1 200 150 5 D2 200 100

3.4 Newton-Raphson based TPNS Simulation Model

As presented in section 3.3, the solution to the unknown pressure and flow

variables are also obtained based on Newton-Raphson solution schemes. Newton-

Raphson technique is complex but powerful for analysis of pipeline network

problems with large number of pipes and compressor stations. The multivariable

Newton-Raphson method which is very important for the analysis of the TPNS can

also be derived following the same procedure as that of the single variable.

A set of nonlinear equations which are obtained from single phase flow

modeling, compressor modeling, looping condition, and mass balance formulations

discussed in section 3.2 can be represented in matrix form.

Let =PN total number of unknown pressure variables.

=QN total number of unknown flow variables.

The total number of unknown variables TotalN is given as:

( )22.3QPTotal NNN +=

The set of single phase pipe flow, looping, compressor, and mass balance

equations can be represented as

73

( )( )

( )

( )23.3

0,,,,,

0,,,,,

0,,,,,

,21,21

,21,212

,21,211

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

=

=

=

••••••

•••

•••

•••

••••••

••••••

QNPNTotalN

QNPN

QNPN

QQQPPPF

QQQPPPF

QQQPPPF

Equation (3.23) can be written in matrix form as [79]

( )24.30~~~

=⎟⎟⎠

⎞⎜⎜⎝

⎛ XF

where the vector ~

X represents the total number of unknown pressure, flow

and temperature variables and ~

F is the corresponding equations generated

from pipe, compressor, mass balance and looping conditions.

The multivariable Newton-Raphson iterative procedure for equation (3.24) takes

the form

( )25.3~~

1

`~

~~~

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡−=

−

old

oldXoldnew XFAXX

where ~A is called Jacobian matrix whose elements are partial derivatives of

the functions with respect to the variables.

The matrix ~A in equation (3.25) is defined as

74

( )26.3

11

2

1

22

1

2

1

1

11

1

1

~

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

∂∂

∂∂

∂∂

∂

∂

•

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

=

••••••

•

•

••••••

••••••

QN

NTotalNTotal

PN

NTotalTotalN

QNPN

QNPN

QF

QF

PF

PF

QF

QF

PF

PF

QF

QF

PF

PF

A

From equation (3.25), the inverse of the Jacobian matrix needs to be computed at

each iterations. However, there is another approach that does not require the rigorous

computation of associated with the inversion of the Jacobian matrix. Equation (3.25)

can be rewritten as[79]

( )27.3~~~~

~

~

⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎥

⎦

⎤⎢⎣

⎡− oldoldnew

oldX

XFXXA

The value of the unknown variables were calculated from equation (3.27)

iteratively until the maximum relative percentage error defined in equation (2.17) is

less than the specified tolerance or the number of iterations reached the desired

value. Figure 3.18 shows the flowchart of the TPNS simulation model based on the

iterative Newton-Raphson solution scheme.

75

Figure 3.18 Flow chart of the TPNS simulation model based on Newton-Raphson

scheme

76

A visual C++ code is developed for the TPNS simulation based on Newton-

Raphson solution technique. In order to handle the visual C++ program efficiently,

the overall code is grouped into several subtasks. These include, subtask for

mathematical formulation, subtasks for matrix elements generation, subtask for input

data, subtask for Gaussian eliminations, subtask for error sorting, subtask for

evaluating the networks, etc. The snapshot of the typical TPNS source code for

analyzing part of Malaysian gas transmission system is shown in Appendix A.

Sample simulation results from the Newton-Raphson based TPNS simulation model

for gunbarrel pipeline network is as shown in Appendix B.

The applications of Newton-Raphson based TPNS simulation model is

demonstrated using three different network configurations. These include gunbarrel,

branched and looped TPNS configurations. These network configurations are the

most widely used in natural gas industry. The application of the Newton-Raphson

based TPNS simulation model also demonstrated using part of the Malaysian gas

transmission network system. The applications of the Newton-Raphson based TPNS

simulations are presented in the following sections.

3.4.1 Case 1B: Gunbarrel Pipeline Network Module

In gunbarrel (linear) pipeline network system where the gas is transported from

source to demand station linearly, there are no junctions and loops involved. As a

result, the basic simulation equations include only flow and compressor equations.

Figure 3.19 shows a typical gunbarrel transmission network system with two

compressor stations. This network was used in order to demonstrate the application

of the TPNS simulation model for the gunbarrel pipeline network system.

77

Figure 3.19 Gunbarrel pipeline network system with two CSs

For the TPNS shown in Figure 3.19, the number of pipes np = 3 and the number

of compressor stations ns = 2. As a result, based on Table 3.3, there will be 4 pressure

variables and 1 flow variable to be determined. There are 3 flow equations and 2

compressor equations available to solve the problem.

From the discussion in section 3.2.1, the pipe flow equations for single phase gas

flow based on general flow equation is summarized as shown in Table 3.11.

Table 3.11 Summary of flow equations for the given gunbarrel TPNS


Start node End node

0 1 201

21

20 QKPP =− 0),( 11 =QPf

2 3 223

23

22 QKPP =− 0),,( 322 =QPPf

4 5 245

25

24 QKPP =− 0),,( 543 =QPPf

Similarly, based on the discussion on compressor stations modeling in section

3.2.3, the remaining compressor equations for the network can be formulated and

represented in functional forms as shown in Table 3.12.

78

Table 3.12 Summary of compressor equations and functional representations

Compressor

stations

Compressor equation Functional representations

CS1 ( ) ( ) ( ) 1]///[ 314

213121

21

1

2 ++++=⎟⎟⎠

⎞⎜⎜⎝

⎛nQAnQAnQAA

ZRTmn

PP

S

m

0),,( 214 =QPPf

CS2 ( ) ( ) ( ) 1]///[ 324

223221

22

3

4 ++++=⎟⎟⎠

⎞⎜⎜⎝

⎛nQAnQAnQAA

ZRTmn

PP

S

m

0),,( 435 =QPPf

Based on the flow equations shown in Table 3.11 and the compressor equations

shown in Table 3.12, the solution to the unknown pressure and flow variables are

obtained using the developed Newton-Raphson based TPNS simulation model.

The application of the Newton-Raphson based TPNS simulation model for the

gunbarrel network was conducted based on nodal pressure requirements and pipe

data shown in Table 3.13 and Table 3.14 , respectively. The pipe data was based on

[47]. The results of the pressure and flow variables and their convergence graphs will

be discussed in Chapter 4.

Table 3.13 Pressure requirements for two CSs and single customer

Node Pressure [kPa]

0 3000

5 4000

Table 3.14 Pipe data for two CSs and single customer

Pipe node

Diameter [mm]


0 1 900 80

2 3 900 80

4 5 900 80

79

3.4.2 Case 2B: Branched Pipeline Network Module

The second type of transmission network analyzed using the developed Newton-

Raphson based TPNS simulation model is branched system. In this system, gas is

transported from the source to different demand stations by branching from the main

source. Figure 3.20 shows typical branched transmission network system with one

compressor station and branching to five customers. In order to demonstrate the

application of the simulation model for the branched pipeline network system, this

network is considered for the analysis.

Figure 3.20 Branched pipeline network system

For this TPNS, the number of pipes np = 10, the number of compressor stations

ns = 1, and the number of junctions nj = 4. As a result, based on Table 3.3, there will

be 6 pressure variables and 9 flow variable to be determined. There are ten flow

equations, one compressor equation and four mass balance equations available to

solve the problem. From the discussion in section 3.2.1, the pipe flow equations for

single phase gas flow based on general flow equation are summarized as shown in

Table 3.15.

80


Pipe node Flow equation Functional representation Start node End node

0 1 2101

21

20 QKPP =− 0),( 111 =QPf

2 3 2123

23

22 QKPP =− 0),,( 1322 =QPPf

3 D1 2113

21

23 CDD QKPP =− 0),( 133 =CQPf

3 4 2234

24

23 QKPP =− 0),,( 2434 =QPPf

4 D2 2224

22

24 CDD QKPP =− 0),( 245 =CQPf

4 5 2345

25

24 QKPP =− 0),,( 3546 =QPPf

5 D3 2335

23

25 CDD QKPP =− 0),( 357 =CQPf

5 6 2456

26

25 QKPP =− 0),,( 4658 =QPPf

6 D4 2446

24

26 CDD QKPP =− 0),( 469 =CQPf

6 D5 2556

25

26 CDD QKPP =− 0),( 5610 =CQPf


( ) ( ) ( ) ( )28.31]///[ 314

213121

2

1

2 ++++=⎟⎟⎠

⎞⎜⎜⎝

⎛nQAnQAnQAA

ZRTmn

PP

S

m


( ) ( )29.30,, 12111 =QPPf

The remaining mass balance equations for the network were formulated and

summarized as shown in Table 3.16.

Table 3.16 Summary of mass balance equations and functional representations

Node Mass balance equation Functional representation

3 121 CQQQ += ( ) 0,, 12112 =CQQQf

4 232 CQQQ += ( ) 0,, 23213 =CQQQf

5 343 CQQQ += ( ) 0,, 34314 =CQQQf

6 544 CC QQQ += ( ) 0,, 54415 =CC QQQf

81

Thus, the solutions to the unknown pressure and flow variables are obtained

using the developed Newton-Raphson based TPNS simulation model based on the

flow equations shown in Table 3.15, compressor equation (3.29) and the mass

balance equations shown in Table 3.16. The application of the Newton-Raphson

based TPNS simulation model for the branched network was conducted based on the

pipe data shown Table 3.17.

Table 3.17 Pipe data for single CS and five customers

Pipe nodes

Diameter [mm]


0 1 900 100

2 3 900 80

3 D1 600 70

3 4 900 80

4 D2 600 90

4 5 900 70

5 D3 600 80

5 6 900 90

6 D4 600 70

6 D5 600 80

3.4.3 Case 3B: Looped Pipeline Network Module

The third type of transmission network system where the developed Newton-

Raphson based TPNS simulation model applied is looped system. In the case of

looped TPNS, gas pipelines diverge and re-converge when gas is transported from

the source to demand stations.

Figure 3.21 shows typical looped transmission network system with one

compressor station for transmitting gas to eight different customers. For this TPNS,

there are 16 pipes, 1 compressor station, 1 loop and 7 junctions. As a result, based on

Table 3.3, there will be 9 pressure variables and 17 flow variable to be determined.

82

There are 16 pipe flow equations, 1 compressor equation, 2 looping equations and 7

mass balance equations which form the 26 independent equations to analyze the

TPNS.

From the discussion in section 3.2.1, the pipe flow equations for single phase gas

flow are formulated and summarized as shown in Table 3.18.

Figure 3.21 Looped pipeline network system

83

Table 3.18 Summary of flow equations and functional representations


0 1 2101

21

20 QKPP =− 0),( 111 =QPf

2 3 2223

23

22 QKPP =− 0),,( 2322 =QPPf

3 4 2334

24

23 QKPP =− 0),,( 3433 =QPPf

4 D1 2114

21

24 CDD QKPP =− 0),( 144 =CQPf

4 D2 2224

22

24 CDD QKPP =− 0),( 245 =CQPf

3 5 2445

25

23 QKPP =− 0),,( 4536 =QPPf

5 D3 2335

23

25 CDD QKPP =− 0),( 357 =CQPf

5 6 2556

26

25 QKPP =− 0),,( 5658 =QPPf

6 D4 2446

24

26 CDD QKPP =− 0),( 469 =CQPf

6 7 2676

27

26 QKPP =− 0),,( 67610 =QPPf

7 D5 2557

25

27 CDD QKPP =− 0),( 5711 =CQPf

7 8 2878

28

27 QKPP =− 0),,( 88712 =QPPf

8 D6 2668

26

28 CDD QKPP =− 0),( 6813 =CQPf

8 9 2989

29

28 QKPP =− 0),,( 99814 =QPPf

9 D7 2779

27

29 CDD QKPP =− 0),( 7915 =CQPf

9 D8 2889

28

29 CDD QKPP =− 0),( 8916 =CQPf

Based on the discussion on sections 3.2.2, the looping condition of the network

can be formulated for the given network as

( )30.3567

2667

556

2556

535

2435

523

2223

5

27

DQl

DQl

DQl

DQl

DQl

l

l +++=

The functional representation for equation (3.30) takes the form

( ) ( )31.30,,,, 7654217 =QQQQQf


84

( ) ( ) ( ) ( )32.31]///[ 314

213121

2

1

2 ++++=⎟⎟⎠

⎞⎜⎜⎝

⎛nQAnQAnQAA

ZRTmn

PP

S

m


( ) ( )33.30,, 12118 =QPPf

The remaining mass balance equations for the network were formulated and


Table 3.19 Summary of mass balance equations and functional representations

Node Mass balance equation Functional representation

2 721 QQQ += ( ) 0,, 72119 =QQQf

3 432 QQQ += ( ) 0,, 43220 =QQQf

4 213 CC QQQ += ( ) 0,, 21321 =CC QQQf

5 354 CQQQ += ( ) 0,, 35422 =CQQQf

6 465 CQQQ += ( ) 0,, 46523 =CQQQf

7 5876 CQQQQ +=+ ( ) 0,,, 587624 =CQQQQf

8 698 CQQQ += ( ) 0,, 69825 =CQQQf

9 879 CC QQQ += ( ) 0,, 87926 =CC QQQf

Based on the flow equations in Table 3.18, looping condition (3.31), the

compressor equation (3.33) and mass balance equations in Table 3.19, the

solutions to the unknown pressure and flow variables were obtained using the

developed Newton-Raphson based TPNS simulation model. The application of the

Newton-Raphson based TPNS simulation model for the looped network was

conducted based on the pipe data shown Table 3.20.

85

Table 3.20 Pipe data for single CS and eight customers

Pipe nodes

Diameter [mm]


0 1 900 80

2 3 900 60

3 4 600 40

4 D1 600 20

4 D2 600 17

3 5 900 55

5 D3 600 18

5 6 900 70

6 D4 600 50

6 7 900 75

7 D5 600 40

7 8 900 184

8 D6 600 18

8 9 900 6

9 D7 200 33

9 D8 900 87

3.4.4 Case Study: Malaysia Gas Transmission System

In this section, the Newton-Raphson based TPNS simulation model was tested

with the actual data from the existing pipeline network system. The network was

identified since it consists of all the pipeline configurations, namely: gunbarrel,

branched, and looped pipeline network systems.

The objective of the case study was to have a comparison between the results of

TPNS simulation model and the actual data collected in the field. However, some

data, like the performance of the compressor and flow consumption by the customers

are considered confidential by the pipeline company. Hence, it would be difficult to

86

make actual comparison between the results of the simulation with that of the field

data. Several options were also tried in order to get the performance of the

compressor.

As an alternative to making an actual comparison between the results of the

TPNS simulation and the field data, the case study was applied to illustrate the

application of the TPNS simulation model to real pipeline network system. The detail

applications of the major subtasks of the TPNS simulation model were evaluated

based on the network system. Sensitivity analysis was also conducted on the basis of

the pipeline configuration system.

The TPNS considered for the case study consists of one compressor station with

two compressors working in parallel and having power capacity of 18 to 24 MW

each. The pipeline network system serves nine major power plant customers and one

Gas District Cooling (GDC) system. The two compressors are working in order to

satisfy the daily requirement of natural gas ranging from 0.8 to 1.7 billion standard

cubic feet per day (BSCFD) which is equivalent to the consumption of gas raging

from 22.65 to 48.14 million metric standard cubic meters per day (MMSCMD). Each

compressor has a capacity of 1 to 1.2 BSCFD which gives a cumulative capacity of

the compressors delivering 2 to 2.4 BSCFD to various customers which is equivalent

to 56.63 to 67.96 MMSCMD.

Due to the limitation of data regarding the performance map of the actual

compressors working in the field, gas pipeline compressors from [55] were used for

the analysis of the network. The compressor has a maximum capacity of 680m3/min

which is equivalent to 40800m3/hr (0.98 MMSCMD). The maximum speed is limited

to 10500rpm and the maximum head of the compressor is 108kJ/kg.

The gas flow rates in the pipe sections are 1Q , 2Q , … etc. and the flow rates

passing out the gas pipeline to various customers are 1CQ , 2CQ ,.. etc. Each customer is

identified by their customer station as 1D , 2D , etc. Figure 3.22 shows the network

87

configuration of part of the Malaysia gas pipeline network system considered for

case study.

There are 19 pipe elements connecting the various nodes with length of pipes

ranges from 6 km to 200 km. The diameter of the pipes ranges from 200 mm to 900

mm. Since the installation of the pipes was done on three different phases, the age of

the pipes is assumed to be ten years for determination of coefficient of friction of the

pipes. This TPNS consists of 1 loop and 8 junctions. As a result, based on Table 3.3,

there are 10 nodal pressures and 20 flow variables to be determined. Therefore, a

total of 30 independent equations were obtained in order to solve the network

problem. There are 19 pipe flow equations, 1 compressor equation, 2 looping

equations and 8 mass balance equations which formed the 30 independent equations

to analyze the TPNS. All the equations were formulated based on the discussion in

section 3.2, page 47. The results for unknown variables were determined on the

basis of multivariable Newton-Raphson algorithm.

88

Figure 3.22 Part of the existing pipeline network system

3.5 Enhanced Newton-Raphson based TPNS Simulation Model

The enhanced Newton-Raphson based TPNS simulation consists of additional

features such as two-phase flow analysis, internal corrosion and temperature

variations. Corrosion and two-phase flow analysis modified the flow equations.

When temperature variation was considered during analysis, it added (np+ ns)

temperature equations with equal number of unknowns. This section discusses two-

phase flow modeling, effect of internal corrosion and temperature variation in TPNS

simulation model.

3.5.1 Two-phase Gas-liquid Flow Modeling

As shown in equation (2.13), the total pressure gradient equation is composed of

three components: frictional, gravitational, and acceleration components. This

89

equation further developed here to enable the calculation of each of the pressure-

gradient components and the total pressure gradient. The basic facts and relationships

between the parameters are developed based on Figure 2.5.

The frictional pressure-gradient component is given by:

( )34.322 2'2'

NS

fmNS

F

Mf

Dvf

DdLdP

ρρ ==⎟

⎠⎞−

where fM is the mixture total mass flux, and mNSf vM ρ= for no-slip condition

and 'f is the fanning friction factor.

For rough pipes, ( )DReff m ε,'' = , and it can be determined from a Moody

chart [23] or from the following equation developed in [70].

( )35.3Re

101021001375.0316

4'

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+×+=

Df ε

For smooth pipes, the friction factor is a function of the Reynolds number only,

( )Reff = , and a Blasius–type equation can be used as given [70]

( ) ( )36.3nReCf −=

where C is constant.

In equation (3.36), for laminar flow, the exponent n is 1 and that the values of

the constant are 16=C and 64=C for fanning and Darcy’s friction, respectively. For

turbulent flow, different values are available for different ranges of the Reynolds

number. For all practical purpose, the correlation covering the widest range of the

Reynolds number is 2.0=n with 046.0=C and 184.0=C for fanning and Darcy’s

friction respectively.

The gravitational pressure gradient component of the pressure drop for two-phase

flow can be determined as

90

( )37.3sinsin θρθρ ggdLdP

NSmG

==⎟⎠⎞−

The acceleration pressure gradient component is given as in [70]:

( )38.31)1(2

2

dLdA

AM

dPdx

dPdx

dLdP

dLdxM

dLdP P

PNS

fLGGLf

A ρυυυ −

⎭⎬⎫

⎩⎨⎧

⎥⎦⎤

⎢⎣⎡ −++=⎟

⎠⎞−

Therefore, the total pressure gradient can be obtained by combining the

frictional, gravitational, and acceleration pressure gradient component given in

equation (3.34), (3.37) and (3.38) respectively, which can be written as:

( )39.31)1(

sin2

22

2'

dLdA

AM

dPdx

dPdx

dLdP

dLdxM

gvfDdL

dP

P

PNS

fLGGLf

NSmNS

ρυυ

υ

θρρ

−⎭⎬⎫

⎩⎨⎧

⎥⎦⎤

⎢⎣⎡ −+++

+=−

Solving for the total pressure gradient from equation (3.39) yields

( )40.3)1(1

1sin2

2

222'

⎥⎦⎤

⎢⎣⎡ −++

−++=−

dPdx

dPdxM

dLdA

AM

dLdxMgvf

DdLdP

LGf

P

PNS

fGLfNSmNS

υυρ

υθρρ

Numerical evaluation

In order to see the effect of each component of the total pressure gradient,

numerical example in [70] is considered here.

The followings are detail of the oil and gas mixture considered for the analysis.

m.D 0510= , 010=θ , skgM L /5.1= , skgM G /015.0= . The temperature of the mixture is

assumed to be 250C and the physical properties of the fluids are: 3/5.1 mkgG =ρ ,3/850 mkgL =ρ , mskgL /100.2 3−×=μ , mskgG /100.2 5−×=μ .

91

Based on the above data and principles of homogeneous two-phase flow analysis,

the frictional pressure gradient can be evaluated based on equation (3.34) as:

mpadLdP

F/3.741=⎟

⎠⎞−

The gravitational pressure gradient can be obtained from equation (3.37) as:

mPadLdP

G/4.219=⎟

⎠⎞−

Similarly, the acceleration pressure gradient component can also be obtained

based on equation (3.38). Note that for constant x , the value of Lυ is also constant.

Furthermore, from the given information the cross-sectional area of the pipe PA is

constant. As a result, the acceleration pressure gradient equation (3.38) reduces to

( )41.32

dLdP

dPdxM

dLdP G

fA

υ=⎟

⎠⎞−

Based on equation (3.41) and assuming ideal gas law, the acceleration pressure

gradient is determined to be

mPadLdP

A/80.19=⎟

⎠⎞−

As a result the total pressure gradient is

mPadLdP /5.1980=⎟

⎠⎞−

Note that for the above homogeneous two-phase gas-liquid mixtures, the

acceleration pressure gradient component contributes only 2.02% to the total

pressure gradient. Hence, the acceleration pressure gradient is usually omitted from

92

calculation. Neglecting the effect of acceleration component of the pressure gradient,

equation (3.40) can be simplified for horizontal pipe flow as

( )42.32 2'mmvf

DdLdP ρ=−

As given in equation (3.36), the frictional coefficient for smooth turbulent flow

can be described as function of Reynolds number as

( )43.3046.0 2.0' −= mRef

The mixture Reynolds number for the two-phase flow is given

( )44.34D

QRem

mmm μπ

ρ=

The mixture velocity can also be expressed for two-phase flow as

( )45.342D

Qv mm π=

Substituting equations (3.43), (3.44), and (3.45) into equation (3.42) results

( )46.344046.02

2

2

2.0

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛×=−

−

DQ

DQ

DdLdP m

m

mmm πμπ

ρρ

By integrating equation (3.46) from 0=L , 1PP = , to LL = , 2PP = we get

( )47.344046.02

2

2

2.0

21 LDQ

DQ

DPP m

m

mmm ⎟

⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛×=−

−

πμπρ

ρ

For two-phase flow mixture, the pressure drop equation for any pipeline element

connecting node i and j shown in Figure 3.2, relating upstream pressure iP ,

downstream pressure jP , and the mixture flow through pipe mQ can be expressed as:

93

( )48.380.1mijji QKPP =−

where ijK is to be determined by the mixture properties and the

characteristics of pipe connecting node i and j .

ijK takes different forms depending on the dimensions of the parameters used

in the flow equations. When ][kPaP , ][kmL , ]/[ 3 hrmQm , ][mmD , ]/[ 3mkgmρ ,

]/[ smKgmμ , the expression for ijK takes the form

( )49.3100791.3 8.4

2.08

DLK

m

mmij

−

⎟⎟⎠

⎞⎜⎜⎝

⎛×=

μρ

ρ

If the pipeline networks system shown in Figure 3.3 is used to transmit two-

phase gas-liquid mixtures, the flow equations governing the simulation can be

generated for the pipes based on equation (3.48). Assuming that the source

pressure at node 0 and the demand pressures at node 6 and 9 are known, the two-

phase flow equations for the pipes and the corresponding functional representation

can be summarized as shown in Table 3.21. Note that for ease of simplification,

the mixture flow rate mQ and single phase gas flow rate Q are represented by

same variable Q .

Table 3.21 Two-phase flow equations and functional representations

Pipe node Two-phase flow equation Functional representation Start node End node

0 1 80.110110 QKPP =− 0),( 111 =QPf

2 3 80.112332 QKPP =− 0),,( 1322 =QPPf

3 4 80.123443 QKPP =− 0),,( 2433 =QPPf

3 7 80.133773 QKPP =− 0),,( 3734 =QPPf

5 6 80.125665 QKPP =− 0),( 255 =QPf

8 9 80.138998 QKPP =− 0),( 386 =QPf

94

The application of the enhanced Newton-Raphson TPNS simulation model for

two-phase flow analysis is demonstrated based on gunbarrel and branched network

configurations.

i) Case 1C: Gunbarrel Pipeline Network Module

When two-phase gas-liquid flows through the gunbarrel TPNS, the flow

equations were affected. It is assumed that only the gas flows to the compressor

station for compression due to severe effect of the liquid to the compressor stations.

In practice, scrubbers are usually installed in front of the compressor station to

remove any unwanted elements.

For the gunbarrel pipeline network system shown in Figure 3.19, the pipes flow

equations for two-phase gas-liquid flow are given based on equation (3.48) and




0 1 80.10110 QKPP =− 0),( 11 =QPf

2 3 80.12332 QKPP =− 0),,( 322 =QPPf

4 5 80.14554 QKPP =− 0),,( 543 =QPPf


3.2.3, the remaining compressor equations for the network can be formulated and



shown in Table 3.12, the solution to the unknown pressure and flow variables were

obtained using the enhanced Newton-Raphson based TPNS simulation model.

95

ii) Case 2C: Branched Pipeline Network Module

The branched pipeline network discussed in 3.4.2 is considered here. Based on

equation (3.48), the pipe flow equations for two-phase gas-liquid flow were

formulated and summarized as shown in Table 3.23.



0 1 80.110110 QKPP =− 0),( 111 =QPf

2 3 80.112332 QKPP =− 0),,( 1322 =QPPf

3 D1 80.11133 CDD QKPP =− 0),( 133 =CQPf

3 4 80.123443 QKPP =− 0),,( 2434 =QPPf

4 D2 80.12244 CDD QKPP =− 0),( 245 =CQPf

4 5 80.134554 QKPP =− 0),,( 3546 =QPPf

5 D3 80.13355 CDD QKPP =− 0),( 357 =CQPf

5 6 80.145665 QKPP =− 0),,( 4658 =QPPf

6 D4 80.14466 CDD QKPP =− 0),( 469 =CQPf

6 D5 80.15566 CDD QKPP =− 0),( 5610 =CQPf

The remaining equations from the compressor maps and mass balance were

formulated following the same procedure as in section 3.4.2. The compressor

equation for the TPNS is given as in equation (3.28) and represented in functional

form as in equation (3.29). The remaining mass balance equations for the network

were formulated and summarized as shown in Table 3.16.

The solutions to the unknown pressure and flow variables were obtained using

the developed enhanced Newton-Raphson based TPNS simulation model based on

the flow equations shown in Table 3.23, compressor equation (3.29) and the mass

balance equations shown in Table 3.16.

96

3.5.2 The Effect of Internal Corrosion

The effect of corrosion is to modify the flow equation during the development of

the governing simulation equations. Since corrosion in oil and gas industries is one

of the serious challenges which affect the performance of the pipeline network

system, the effect of corrosion on the performance of the gas transmission system

have to be analyzed and incorporated during the development of TPNS simulation

model.

So far, limited information is available from the literatures about how the

roughness of the pipes varies with the age of the pipe. In this section, a solution

scheme is developed to incorporate the age of the pipe to flow equation. Figure 3.23

shows the general procedure proposed to incorporate the effect of corrosion with age

of the pipe and flow modification.

Data of roughness of the coated pipe with service life of the pipe from [22] was

used for developing the correlation between the age of the pipe and roughness.

Several options were tried in order to get the best fit for the data of pipe roughness

against the service life of the pipe. It is observed that, even if the effect of pipe

roughness increase with increase in age of the pipe, the best fit for the data was

obtained when the approximation was done with 6 degree polynomial function with

correlation coefficient of 88.02 =R . However, this approximation gave negative

roughness values when the age of the pipe increased beyond the data points. A better

approximation for the given data was obtained when the data is approximated with

exponential function with correlation coefficient of 592.02 =R .

97

Figure 3.23 Proposed scheme for incorporating the age of the pipes in flow equations

Based on the data points, exponential function for approximating the relationship

between the roughness of the pipe with that of the age of the pipe is obtained to be:

( )50.300353.0 03802.0 yer =

where r is the roughness of the pipe in mm and y is the age of the pipe in year.

Most gas pipelines operate in the turbulent zone. For the fully turbulent zone,

American Gas Association (AGA) recommends using the following formula for

calculating the transmission factor F which is based on the relative roughness Dr

and independent of the Reynolds number [23] .

( )51.37.34 10 ⎟⎠⎞

⎜⎝⎛=

rDLogF

where F is the transmission factor, r is the roughness of the pipe and D is

the diameter of the pipe.

The relationships between the transmission factor F and friction factor f can be

express as [23]:

98

( )52.32f

F =

Substituting equation (3.52) in equation (3.51) and rearranging gives:

( )53.37.32

12

10 ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

=

rDLog

f

From equations (3.50) and (3.53), it can be seen that the friction factor is a

function of the age of the pipe. Figure 3.24 shows the effect of the years in service of

the pipe on friction factor for various diameters of the pipe.

Figure 3.24 The effect of length of service of the pipe on friction factor

The general flow equation for a single phase gas flow in equation (3.1) can be

modified based on the friction factor in equation (3.53) in order to incorporate the

effect of the age of the pipe on the pressure drop of the TPNS. For any pipe (Figure

3.2) connecting the upstream node i, downstream node j, and the flow through pipe

99

Q , the single phase flow equation with corrosion relating the two nodes can be

represented as

( )54.32'22 QfKPP ijji =−

where

( )55.3103599.42

58' L

TP

DGZTK

n

nij ⎟⎟

⎠

⎞⎜⎜⎝

⎛×=

If all the pressure and flow variables are unknown, equation (3.54) can also be

represented in functional form as

( )56.30),,( =QPPf ji

The application of the enhanced Newton-Raphson based TPNS simulation model

for incorporating the effect of internal corrosion is demonstrated based on gunbarrel

and branched network configurations.


When corrosion is considered during the analysis of gas transmission system, the

overall effect is to modify the pressure drop equations. The gunbarrel transmission

network configuration shown in Figure 3.19 is considered here. For this pipeline

network, based on equation (3.54), the pipe flow equations for single phase gas flow

with corrosion were formulated and summarized as shown in Table 3.24.



0 1 2101

21

20 QfKPP =− 0),( 11 =QPf

2 3 2223

23

22 QfKPP =− 0),,( 322 =QPPf

4 5 2345

25

24 QfKPP =− 0),,( 543 =QPPf

100


3.2.3, the remaining compressor equations for the network were formulated and



shown in Table 3.12, the solution to the unknown pressure and flow variables were

obtained using the enhanced Newton-Raphson based TPNS simulation model.


For the branched pipeline network shown in Figure 3.20, based on equation

(3.54), the pipe flow equations for single phase gas flow with corrosion were

formulated and summarized as shown in Table 3.25.


considering corrosion


0 1 21101

21

20 QfKPP =− 0),( 111 =QPf

2 3 21223

23

22 QfKPP =− 0),,( 1322 =QPPf

3 D1 21313

223 CDD QfKPP =− 0),( 133 =CQPf

3 4 22434

24

23 QfKPP =− 0),,( 2434 =QPPf

4 D2 22524

224 CDD QfKPP =− 0),( 245 =CQPf

4 5 23645

25

24 QfKPP =− 0),,( 3546 =QPPf

5 D3 23735

225 CDD QfKPP =− 0),( 357 =CQPf

5 6 24856

26

25 QfKPP =− 0),,( 4658 =QPPf

6 D4 24946

226 CDD QfKPP =− 0),( 469 =CQPf

6 D5 251056

226 CDD QfKPP =− 0),( 5610 =CQPf

The remaining equations from the compressor maps and mass balance were

formulated following the same procedure as in section 3.4.2. The compressor

101

equation for the TPNS is given as in equation (3.28) and represented in functional

form as in equation (3.29). The remaining mass balance equations for the network

were formulated and summarized as shown in Table 3.16.

The solutions to the unknown pressure and flow variables were obtained using

the developed enhanced Newton-Raphson based TPNS simulation model based on

the flow equations shown in Table 3.25, compressor equation (3.29) and the mass

balance equations shown in Table 3.16.

3.5.3 Modeling Temperature Variations

When the temperature of the gas varies due to heat transfer and compression, it is

essential to consider temperature variations during modeling. As presented in section

2.4.2, the variations of temperature within the TPNS are described based on

equations (2.14) and (2.15). These two equations should have to be integrated into

the governing simulation equations to represent the effect of temperature variations

for the enhanced Newton-Raphson based TPNS simulation.

Equation (2.14) is modified before it is integrated into the governing simulation

equations as the governing simulation equation takes only equations which are

functions of pressure, flow, and temperature variables. The mass flow rate M used

in equation (2.15) is given as function of volumetric flow Q and density of the gas as

( )57.3QM ρ=

Substituting equation (3.57) in equation (2.15) yields

( )58.3QCLUD

Pρπθ Δ

=

Therefore, equation (2.14) is modified for any pipe element (Figure 3.2)

connecting the upstream node i, downstream node j, and the flow through pipe Q , as

102

( )59.3)( /QTijsisj eTTTT −+=

where PC

LUDTijρ

π Δ=

If all the variables in equation (3.59) are unknown, it can also be represented in

terms of function form as

( )60.30),,( 21 =QTTf

The second equation to represent the variation of temperature in gas transmission

network system is equation (2.16). Since equation (2.16) is function of the required

variables, it was used without any modifications on it. If all the variables are

unknown in equation (2.16), the functional form of the equation takes the form

( )61.30),,,( 2121 =PPTTf

Equation (3.60) and (3.61) are the basic equations which govern the enhanced

Newton-Raphson based TPNS simulation under variable temperature conditions.

The application of the enhanced Newton-Raphson TPNS simulation model for

temperature variations is demonstrated based on the gunbarrel pipeline network

configurations shown in Figure 3.19. If the temperature of the gas is assumed to be

constant, there were 4 pressure variables and 1 flow variable to be determined for the

TPNS. When temperature variation is considered during the analysis, np+ns temperature equations with equal number of unknown temperature variables were

added to the governing simulation equations. As a result, there were five more

temperature equations and five unknown temperature variables introduced to the

system. The numbers of total unknown pressure, flow, and temperature variables

became ten.

Single phase flow model was used for the flow analysis. The summary of the

pipe flow equations for single phase gas flow model using general equation was

103

given as in Table 3.11. The remaining compressor equations for the network were

formulated and represented in functional forms as shown in Table 3.12.

The temperature equations were developed and represented in functional form

for the given pipeline network as shown in Table 3.26.

Table 3.26 Summary of temperature equations for gunbarrel TPNS

Temperature equation Functional representation

QTss eTTTT /01

01 )( −+= ( ) 0,16 =QTf

kk

PP

ZZ

TT

1

1

2

2

1

1

2

−

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛ ( ) 0,,, 21217 =PPTTf

QTss eTTTT /23

23 )( −+= ( ) 0,, 328 =QTTf

kk

PP

ZZ

TT

1

3

4

2

1

3

4

−

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛ ( ) 0,,, 43439 =PPTTf

QTss eTTTT /45

45 )( −+= ( ) 0,, 5410 =QTTf

Therefore, based on the flow equations in Table 3.11 , the compressor equations

in Table 3.12, and temperature equations in Table 3.26, the solutions to the unknown

pressure, flow, and temperature variables were obtained using the enhanced NR

based TPNS simulation model.

3.6 Performance Evaluation of the TPNS Simulation Model

As it is observed from the solution schemes discussed in the previous sections,

the successive substitution and Newton-Raphson methods provided solutions to the

unknown pressure, flow and temperature variables which are essential for evaluating

the performance of TPNS. The performance of the TPNS includes evaluation of the

flow capacities through each pipe elements, compression ratios at each compressor

stations, and the overall power consumption of the TPNS.

104

The flow capacities through each pipe are determined directly from the value of

the unknown variables obtained using either of the solution schemes discussed

above. The compression ratios (CR) at each compressor stations are evaluated based

on the pressure variables obtained by using the solution schemes. Based on the nodal

pressures, the compression ratio is defined as the ratio of discharge pressure to

suction pressure.

Based on the pressure and flow variables obtained, the simulation model is used

to evaluate the energy consumption of the TPNS. The amount of energy input to the

gas by the compressors is dependent upon the pressure of the gas and flow rate. The

power required by the compressor that takes into account the compressibility of gas

is given as [23]

( )62.31121

0639.4121

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛

−=

−kk

s

d

aS P

PZZQTk

kHPη

where HP is the compression power in kW.

3.7 Summary

The basic equations for the TPNS simulation are derived from the principles of

flow of fluid through pipe, compressor characteristics and the principles of mass

balance at the junction of the network. The existences of loops within the TPNS

create additional equations to the simulation model. These equations are developed

from the principle of looping conditions which states the pressure drop along any

closed loop is zero.

The solution for the unknown variables could be determined on the basis of an

iterative successive substitution or Newton-Raphson schemes. The method of

successive substitution is closely associated with the IFD where the governing

105

simulation equations are arranged in such a way that only one output is obtained

from the blocks. On the other hand, the Newton-Raphson solution scheme is

complex but powerful for the analysis of TPNS where the number of equations is

large.

The modeling of the application of iterative successive substitution scheme based

TPNS simulation model is demonstrated using two network configurations. The

development of the basic governing equations and the corresponding IFD generation

were discussed for the networks considered. The application of the Newton-Raphson

based TPNS simulation is also demonstrated using the three most commonly

networks configurations, namely: gunbarrel, branched and looped.

The next chapter discusses the results obtained by testing the developed TPNS

simulation model for different TPNS configurations. Table 3.27 shows the summary

of the different cases considered for the analysis.

106

Table 3.27 Various cases considered for the analysis using developed TPNS

simulation model

Model Case Module Section

Successive

substitution based

TPNS simulation

model

Case 1A 1CS 2 Customers 4.2.1

Case 2A 2CSs 2 Customers 4.2.2

Newton-Raphson

based TPNS

simulation model

Case 1B Gunbarrel 4.3.1

Case 2B Branched 4.3.2

Case 3B Looped 4.3.3

Malaysia TPNS

General 3.4.4

Enhanced Newton-

Raphson based TPNS

simulation model

Case 1C Gunbarrel 4.4.1

Case 2C Branched 4.4.1

Case 3C Gunbarrel 4.4.2

Case 4C Branched 4.4.2

The enhanced Newton-Raphson based TPNS simulation consists of additional

features such as two-phase flow analysis, internal corrosion and temperature

variations. Corrosion and two-phase flow analysis modified the flow equations.

When temperature variation is considered during analysis, it added temperature

equations with equal number of unknowns.

107

CHAPTER 4

RESULTS AND DISCUSSIONS

4.1 Introduction

In this chapter, the results of detail application of the TPNS simulation model for

various pipeline configurations are presented. The first part discusses the results of

the application of the TPNS simulation model based on successive substitution

solution scheme. The second part discusses the results of the application of the TPNS

simulation model based on the iterative Newton-Raphson solution scheme. The

application of the TPNS simulation model for real pipeline network system is also

discussed in this section. The third part discusses the results of enhanced Newton-

Raphson based TPNS simulation model. The fourth part of this Chapter discusses the

verification and validation of the simulation model.

4.2 Successive Substitution based TPNS Simulation Model

The results of the applications of successive substitution solution scheme for

TPNS simulation are demonstrated based on two different pipeline configurations

discussed in section 3.3.1 and 3.3.2. The successive substitution based simulation

model was implemented using excel based spreadsheet. The results of the analysis

were reported by Woldeyohannes and A. Majid [87].

108

4.2.1 Case 1A: Single Compressor Station and Two Customers Module

The single compressor station with two customers’ network shown in Figure 3.14

on page 66 is considered here for the analysis. Using the procedures described in

section 3.3.1, the results of the pressure and flow variables for this TPNS for the first

50 iterations are presented in Appendix C.

The convergences of the pressure and flow variables were studied. As stated in

[81], the calculation sequence dictated by the structure of the IFD determines

whether the sequence will converge or diverge. P3 and Q1 were the variables

identified from the IFD shown in Figure 3.15 on page 68 to start the analysis. When

the initial estimations for P3 and Q1 varied, there were two phenomena observed from

the simulation. The first case was the divergence case. In this case, the relative errors

calculated from each successive iterations followed irregular trends. The second case

was the slow convergence rate. In this case, when the initial estimations were far

from the final solutions of the variables, it took longer iterations to arrive the final

solution.

The convergence of the pressure variables for the TPNS is as shown in Figure

4.1. The pressure variables converged to their final solution with different rate of

convergence. The relative errors of the pressure variables started to become stable at

the 17th iteration for P2 and 16th iterations for P1 and P3.

109

Figure 4.1 Convergence of pressure variables (case 1A)

The convergences of the flow variables are shown in Figure 4.2. Around 20

iterations were required to get stable convergence graph for the flow variables.

However, QC1 and QC2 became stable starting from the 16th iterations as the initial

assumed values are near to these variables compared to Q1. The negative flow

observed for QC1 for the two iterations was due to the high initial estimation for P3.

110

Figure 4.2 The convergence of flow variables (Case 1A)

From the convergence graphs (Figure 4.1 and Figure 4.2), it is observed that, the

maximum relative percentage error after the 20th iterations is 1.21%. At the end of

the 50th iterations the maximum relative percentage error is 1.33E-04.

4.2.2 Case 2A: Two Compressor Stations and Two Customers Module

The two compressor stations two customers network shown in Figure 3.16 on

page 69 is considered here for the analysis. Using the procedures described in section

3.3.2, the results of the pressure and flow variables for this TPNS for the first 200

iterations.

The convergences of the pressure and flow variables were also studied. Figure

4.3 shows the convergence of the pressure variables for the first 200 iterations. The

corresponding convergences of the flow variables are shown in Figure 4.4. After the

20th iterations, the maximum relative percentage error observed was 32.7%. At the

end of the 50th iterations the maximum relative percentage error is 8.15E-02. These

111

show that as the number of unknown variables increased, the successive substitution

scheme took longer time to arrive at the specified tolerance as presented in [81].

Figure 4.3 The convergence of pressure variables (case 2A)

112

Figure 4.4 The convergence of flow variables (case 2A)

4.2.3 Limitations of Successive Substitution TPNS Simulation Model

Results from the numerical evaluation of the networks shown in Figure 3.14 on

page 66 and Figure 3.16 on page 69 revealed that the successive substitution based

TPNS simulation model could be applied to determine the flow and pressure

variables. However, it was observed that it took 50 iterations for case 1A and 200

iterations for case 2A to converge to relative percentage of errors of 1.33x10-4 and

8.15x10-4, respectively. As the number of pressure and flow variables increased, it

required more iteration to converge to the final solutions. Comparison of successive

substitution based TPNS simulation with that of Newton-Raphson based TPNS

simulation is reported by Majid and Woldeyohannes [88] based on two different

network configurations.

Although successive substitution scheme is straight forward technique and is

usually easy to program, sometimes the method may result in very slow convergence

rate depending on the IFD and the initial estimation for the unknown variables.

113

Hence, the method is limited only for TPNS with few numbers of pipes, compressor

stations and junctions. In addition, it is difficult to get the IFD of the given pipeline

network as the number of branches increases. Several correct IFD can be developed

for the given pipeline network system. However, it is very difficult to identify the

IFD which is convergent as discussed in [81]. For instance, for the TPNS shown in

Figure 4.5, one of the correct IFD is shown in Figure 4.6. It required testing many

alternatives to arrive at the correct IFD.

Figure 4.5 Pipeline network with three CSs serving four customers

The TPNS consists of 10 pipes, 3 compressor stations, 3 junction, and with no

loop. Therefore, np = 10, ns = 3, nl = 0, and nj = 3. As a result, there are 9 nodal

pressures and 7 flow parameters to be determined. Therefore, a total of 16

independent equations have to be obtained in order to solve the network problem.

The IFD is developed based on the governing simulation equations and shown in

Figure 4.6.

114

Figure 4.6 IFD for pipeline network with three CSs and four customers

4.3 Newton-Raphson based TPNS Simulation Model

The multivariable Newton-Raphson method which is discussed in section 3.4 on

page 72 is used for the analysis of TPNS with large number of pipes, compressor

stations, branches and loops. This section discusses results of detailed application of

the Newton-Raphson based TPNS simulation model using gunbarrel, branched and

looped configurations.

115

4.3.1 Case 1B: Gunbarrel Pipeline Network Module

The results and discussions presented in this section are based on the gunbarrel

TPNS network configurations shown in Figure 3.19 on page 77 and the methodology

presented in section 3.4.1 on page 76.

The network was analyzed with the aid of the developed Newton-Raphson based

TPNS simulation model. Gas pipeline compressors from [55] were used for the

analysis. The speed of the compressor was 8000 rpm. Note that the analysis could

also be done based on various speeds as long as the compressor speed is within the

operational limit. In order to make the multivariable Newton-Raphson convenient for

application, only single estimation that was used for all unknown variables was

required to start the iterations. The results for the unknown pressure variables and

flow variable were obtained with less than ten iterations. Table 4.1 shows the results

of the unknown nodal pressures and flow variable for the first ten iterations. Note

that pressure is measured in kPa and flow rate is measured in m3/hr.

Table 4.1 Results of nodal pressures and flow variables (case 1B)

Iteration P1 P2 P3 P4 Q Max. error 0 3000.00 3000.00 3000.00 3000.00 3000.00

1 2982.71 3591.98 3574.69 4183.96 2.40E+06 0.99875

2 1308.05 3665.21 2253.81 5210.26 1.49E+06 0.60892

3 2109.82 3691.36 2997.95 4638.76 987089 0.510994

4 2438.82 3545.31 3064.63 4384.63 715543 0.379496

5 2471.25 3505.5 3064.8 4346.88 638202 0.121187

6 2472.77 3505.01 3065.89 4345.74 632588 0.008874

7 2472.77 3505.01 3065.9 4345.73 632559 4.56E-05

8 2472.77 3505.01 3065.9 4345.73 632559 1.2E-09

9 2472.77 3505.01 3065.9 4345.73 632559 9.94E-16

10 2472.77 3505.01 3065.9 4345.73 632559 1.13E-16

116

The convergences of the pressure and flow variables were studied. Based on the

results obtained in Table 4.1, the convergence of nodal pressures and flow variables

for the network system are shown in Figure 4.7 and Figure 4.8, respectively.

Convergence was achieved in most of the initial estimations attempted for the

Newton-Raphson based TPNS simulation model. An initial estimation near to the

demand pressure requirements for both pressure and flow variables gave

convergences for the attempted simulation runs.

The convergence graphs shown in Figure 4.7 and Figure 4.8 indicated that the

relative percentage errors became stable starting from the 4th iteration. The high

value for Q that is observed at the second iteration in Figure 4.8 is due to the value of

initial estimation which is far off compared to the final solution.

117

Figure 4.7 Convergence of pressure variables (case 1B)

Figure 4.8 Convergence of flow variable (case 1B)

118

The results of Newton-Raphson based TPNS simulation model for the gunbarrel

network configurations are compared with the method proposed by Wu et al [47].

The pressure range for each node is between 689.5 kPa and 6895 kPa. There are 5

compressors within each compressor stations. The flow through pipes is assumed to

be 600 million standard cubic feet per day (MMSCFD) or equivalent to 707,921

million metric standard cubic meters per day (MMSCMD). The problem for the

network is the determination of nodal pressures which minimizes the fuel cost of the

compressors.

Wu et al [47] determined solutions for the optimal nodal pressures for

minimizing the fuel consumption of the system using exhaustive search method. The

results of the nodal pressures are shown in Table 4.2.

Table 4.2 Results of nodal pressures [47]

Node Pressure [kPa]

0 4112.88

1 3433.59

2 3640.43

3 2850.70

4 3088.85

5 2101.13

The pipeline network was also analyzed using the TPNS simulation model. In

TPNS simulation model, it is assumed that maintaining pressure requirement is

critical and therefore, the source pressure P0 and the demand pressure P5 are assumed

to be known. Therefore, the problem in TPNS simulation is finding the remaining

nodal pressures, flow rate, number of compressors within the stations, power

consumption and the speed of the compressors which satisfies the requirement.

A compressor map from the existing pipeline network system [89] was used for

the analysis. The characteristic map of the compressor was available in terms of the

119

discharge pressure versus flow capacity. As a result, the discharge pressure versus

flow capacity curves were approximated based on three degree polynomials.

For each compressor stations, identical compressors working with the same

speed are assumed. The simulation experiments were conducted by varying the

number of compressors and speed of the compressors in order to study the flow

capacity and the nodal pressures. From the simulation experiments, it was observed

that the minimum number of compressors required to meet the flow requirement was

8. Several speeds of compressor were tested for meeting the customer specifications.

The compressor speeds starting from 5550 rpm satisfied the requirement with various

power consumptions, nodal pressures and compression ratios.

It was observed that as the speed of the compressors increased, the results of

nodal pressures from the TPNS simulation were getting closer to the results of nodal

pressures obtained by Wu et al [47]. However, an increase in deviation of flow

parameter was observed when speed of compressors increased. After conducting

simulation experiments, compressor speed of 5775 rpm gave better results of nodal

pressures with maximum percentage error of 2.37% which was obtained at node 5 of

the network. The corresponding flow deviation was 10.71%. Figure 4.9 shows

comparison of the result of nodal pressures obtained from the TPNS simulation

model at speed of 5775 rpm and the results obtained in [47].

120

Figure 4.9 Comparison of nodal pressures

Wu et al [47] obtained the solution for nodal pressures so that the fuel

consumption for the system is minimized. The results obtained from TPNS

simulation model using the Newton-Raphson shows that the simulation model could

be used to compare various operations of the compressor. This could help to compare

the alternatives and select the one with minimum power consumption.

4.3.2 Case 2B: Branched Pipeline Network Module

The results and discussions presented in this section are based on the branched

TPNS network configurations shown in Figure 3.20 on page 79 and the methodology

presented in section 3.4.2 on page 79. The analysis was done based on source

pressure of 3500kPa and demand pressure requirement 4500kPa. The results and

analyses on the network were reported by Woldeyohannes and Majid [90].

121

The network was analyzed using TPNS simulation model based on Newton-

Raphson solution method. Two compressors were working for the system. The

characteristics map of the compressors were taken from Kurz and Ohanian [59]. For

the specified condition, it was assumed that the speed of the compressor is 8800 rpm.

Note that the analysis could also be done at any arbitrary speed within the working

limits of compressors. The results for the unknown pressure variables and flow

variables were obtained with less than ten iterations with relative percentage error of

1.76979E-15. Figure 4.10 shows the convergence of some of the nodal pressures to

the final pressure solutions for the first ten iterations. The convergences of the

remaining nodal pressure variables followed the same trends. The convergence of the

corresponding flow parameters are shown in Figure 4.11 and Figure 4.12.

122

Figure 4.10 Convergence of pressure (case 2B)

Figure 4.11 Convergence of main flow variables (case 2B)

123

Figure 4.12 Convergence of lateral flow variables (case 2B)

4.3.3 Case 3B: Looped Pipeline Network Module

The results and discussions presented in this section are based on the looped

TPNS network configurations shown in Figure 3.21 on page 82 and the

methodology presented in section 3.4.3 on page 81. The network was analyzed based

on single phase gas flow analysis. The pressure at node 0 was 3500kPa and the

demand pressure from the customer was assumed to be 4000kPa. The results of the

analysis using TPNS simulation for looped network configuration were reported by

Woldeyohannes and Majid [91].

124

The rate of convergence to the final solution depends on the initial estimation

and usually high at the beginning of the iterations. Based on the pipe data shown in

Table 3.20 and gas compressor data from [55], the results of the unknown pressure

and flow variables were obtained for the first ten iterations. The analysis was

performed based on compressor speed of 8500 rpm. For the initial estimations of

4000kPa for unknown pressure variables and 4000m3/hr for unknown flow

variables, the results from the TPNS simulation model using Newton-Raphson

method at the end of the 10th iterations are shown in Table 4.3 and Table 4.4. The

corresponding compression ratio and power consumption for the system were

obtained to be 1.49837 and 14.1147 MW, respectively. At the end of the 10th

iteration, the maximum percentage relative error was 7.197x10-11.

Table 4.3 Results of nodal pressures after ten iterations

Node Pressure [kPa]

0 3500.00

1 2880.47

2 4316.02

3 4136.33

4 4014.28

5 4064.4

6 4047.89

7 4046.37

8 4006.1

9 4005.68

125

Table 4.4 Results of main and branch flow variables after 10 iterations

Main flow [m3/hr] Branch flow [m3/hr]

Q1 740381 QC1 88740.9

Q2 529934 QC2 96253

Q3 184994 QC3 199260

Q4 344940 QC4 102997

Q5 145679 QC5 113299

Q6 42682.8 QC6 61105.3

Q7 210447 QC7 2564.04

Q8 139831 QC8 76161.8

Q9 78725.9 - -

The convergences of the pressure variables at node 1 and 2 and the main flow

variable 1Q were studied under various initial estimations. Figure 4.13 and Figure

3.14 show the convergence of nodal pressures to the final pressure solutions for the

first ten iterations at node 1 and 2 for initial estimations of 3000, 4000 and 10000kPa.

The convergences of the remaining nodal pressures followed the same trend as that

of the pressure at node 2.

126


(case 3B)

Figure 4.14 Convergence of P2 for the first ten iterations for the initial estimation

(case 3B)

127

The convergence of the main flow variable 1Q for the initial estimations of 3000,

4000 and 10000 m3/hr is shown in Figure 4.15. The simulation model produced

solutions for the unknown variables with wide range of initial estimation.


(case 3B)

The results of the TPNS simulation model for single phase gas flow analysis

were compared with the method proposed by Osiadacz [7] based on the looped

network configuration shown in Figure 4.16. The pipeline details and the demand

requirements for the TPNS are shown in Table 4.5 and Table 4.6, respectively. For

this problem, Osiadacz applied Newton loop-node method in order to get the flow

and pressure variables by assuming fixed pressure ratios for each compressor.

However, the final nodal pressures obtained after achieving the predefined error

limits failed to satisfy the previously assumed pressure ratios. For instance, the

128

pressure ratio at CS1 was assumed to be 1.8 and that of CS2 was assumed to be 1.4.

However, the pressure ratios after the solutions were obtained were actually 1.34 and

1.1832 for CS1 and CS2, respectively.

Figure 4.16 Pipeline networks with 10 pipes and two compressor stations [7]

Table 4.5 Pipe data [7]

Pipe nodes

Diameter [mm]


1 2 700.00 70.00

1 3 700.00 60.00

2 3 700.00 90.00

2 4 600.00 50.00

3 6 600.00 45.00

5 8 600.00 70.00

7 9 600.00 80.00

8 9 500.00 70.00

8 10 500.00 45.00

9 10 500.00 75.00

129

Table 4.6 Pressure and flow requirements [7]

Node Flow [m3/hr] Pressure [kPa]

1 - 5000

2 20000 -

3 20000 -

4 0 -

5 0 -

6 0 -

7 0 -

8 15000 -

9 30000 -

10 45000 -

The simulation experiments were conducted by varying the number of

compressors and speed of compressors. The analysis was conducted using the

performance characteristics of the compressors taken from [59]. Similar compressors

are assumed to work on both compressor stations. From the simulation experiments,

it was observed that one compressor for each compressor station is sufficient to meet

the customer requirements.

Simulation experiments based on the given compressors characteristics showed

that, the two compressor stations have to work nearly with the same compression

ratios. An increase in compression ratio for the first station could result more flow

through pipe 1 and reversal flow through pipe 3. This might cause flow reversal in

the second compressor station. For instance, the compressor ratio mentioned for CS2

by Osiadacz [7] was achieved when the compressor runs with speed of 4800rpm.

Based on this reference speed for CS1, the speed of the compressor increased to

improve the compression ratio of CS2. The compressor speed at CS1 was increased

till flow reversal starts to CS (i.e. 5450 rpm). It was observed that, the deviations of

the nodal pressures and flow variables increased as the speed increase towards the

speed of 5450 rpm.

130

After conducting simulation analyses based on the requirements, compressors

speeds of 5025 rpm for CS1 and 4750 rpm for CS2 gave results of nodal pressures

and flow variables close to Osiadacz [7]. Mean absolute percent error of 5.10% was

observed between the two methods. The variations of the flow and nodal pressure

variables could be as a result of the type of flow equations that were used in the

analysis and the oversimplification of compressor stations in the case of the method

in [7]. Panhandle ‘A’ flow equation was used in [7] where as general flow equation

has been applied in the developed TPNS simulation. As developed in section 3.2.3,

the TPNS simulation model consists of detailed characteristics of the compressor

rather than only limited to compression ratio.

The results of comparison of the TPNS simulation model and that of Osiadacz

[7] for flow rate through pipes is shown in Figure 4.17. From the figure, it is

observed that the maximum deviation between the results of TPNS simulation model

and the results from Osiadacz [7] occurred at pipe 1, 4 and 6. This is as a result of the

higher compression ratio assumed at CS1 which was far from the calculated value.

The negative flow observed in pipe 3 of the network showed that the actual flow

direction for the gas is opposite to the assumed direction. Hence, the actual flow in

pipe 3 is from node 3 to node 2.

131

Figure 4.17 Comparison of flow rates between TPNS simulation and results of

Osiadacz [7]

The comparison of the corresponding nodal pressures between the two methods

is shown in Figure 4.18. From the figure, it is observed that higher deviation at nodal

pressures between the Osiadacz [7] and the results from the developed TPNS

simulation model happened at node 4. This is as a result of the value of the CR

assumed at CS1. Generally, the results of nodal pressure from the TPNS simulation

model is higher than the nodal pressures obtained from Osiadacz [7]. This could be

as a result of the flow equations used in both methods. The general flow equation

which was used in TPNS simulation model result less pressure drop compared to the

Panhandle ‘A’ flow equation which was used by Osiadacz [7].

132

Figure 4.18 Comparison of nodal pressures between TPNS simulation and results of

Osiadacz [7]

4.3.4 The Malaysia Gas Transmission Network Case Study

In this section, the TPNS simulation model using the Newton-Raphson method

was tested with the actual data from the existing pipeline network system. The

network was identified since it consists of all the pipeline configurations, namely:

gunbarrel, branched, and looped pipeline network systems.

Based on the data from the existing pipeline network system presented in section

3.4.4 on page 85, the TPNS simulation was evaluated. The various subtasks of the

simulation model were evaluated on the basis of the data taken from the field. The

main modules of the TPNS simulation model include input parameter analysis,

function evaluation module, and network evaluation module. Single phase flow

analysis at constant gas flow temperature is assumed for the evaluation of the TPNS

simulation model with the existing pipeline network system.

133

i) Input Parameter Analysis

This phase of the TPNS simulation consists of taking the input from the user and

making analysis in order to make the data suitable for the next phase of the

simulation. The input to the simulation includes pipe data, compressors data and

customer requirements.

a) Pipe Input Data

The TPNS under consideration for the case study consists of 19 pipes with

various diameter and length. The first step of the simulation is to take the relevant

input regarding the number of pipes involved, diameter of the pipes, length of the

pipes and age of the pipes. Using the input information, the TPNS simulation

analyzes the pipe data to determine the friction factor of each pipes as well as the

pipe flow resistance ijK which was defined in equation (3.1) on page 47. Based on

the data of the pipeline network system under consideration, the flow resistance and

friction factor for each pipe were determined and shown in Table 4.7.

134

Table 4.7 Results of input analysis for pipes

Pipe nodes Friction factor f

Flow resistance

ijKStart node End node

0 1 0.007407 7.62395E-06

2 3 0.007407 5.71796E-06

3 D1 0.007878 4.61777E-05

3 D2 0.007878 4.61777E-05

2 4 0.007407 5.71796E-06

4 5 0.007878 3.07851E-05

5 D3 0.007878 1.53926E-05

5 D4 0.007878 1.30837E-05

4 6 0.007407 5.24147E-06

6 D5 0.007878 1.38533E-05

6 7 0.007407 6.67096E-06

7 D6 0.007878 3.84814E-05

7 8 0.007407 7.14746E-06

8 D7 0.007878 3.07851E-05

8 9 0.007407 1.75351E-05

9 D8 0.007878 1.38533E-05

9 10 0.007407 5.71796E-07

10 D9 0.009403 0.00736663

10 11 0.007407 8.29105E-06

b) Compressor Input Data

After the preprocessing of the pipeline information has been completed, the next

step of the TPNS simulation is to take the data related to compressor station as an

input. The data related to compressors consists of the performance map of the

compressors, number of compressors working within the station and the compressor

speed. Based on the performance map of the compressor used, the Newton-Raphson

based TPNS simulation model carried out the analysis to determine the coefficient of

135

the mathematical equation which represents the characteristics of the compressor.

The coefficients were determined as discussed in section 3.2.3 on page 52.

c) Customer Requirement

One of the basic inputs for the TPNS simulation model is to take data related to

the pressure requirement of the customers. The customer specification in terms of

pressure requirement is usually given to satisfy the remote customer pressure

requirement. This is because, if the pressure at the end point is satisfied, the entire

system will have sufficient pressure requirement. For the case study under

consideration, the maximum pressure at the end point of the TPNS could reach

6800kPa and the minimum pressure is limited 4000kPa. The pressures at the end of

the pipe joining the customer with the system will be regulated to the customer

requirement. The source pressure varies from 3000kPa to 3500kPa.

d) Initial Estimations and Maximum Error Limit

The TPNS simulation requires an initial estimation of the unknown variables.

The relative error introduced depends on the initial estimation of the unknown

variables. The simulation model was tested by giving wide range of initial

estimations for the unknown variables and convergences were achieved in most of

the trails. Proper estimation for the nodal pressures could be obtained from the exit

pressure requirements at the various demand stations. Furthermore, proper estimation

for the flow variables could be obtained from the performance characteristics of the

compressor provided for the simulation.

ii) Output of TPNS Simulation

The final results from the TPNS simulation for the required nodal pressure and

flow through pipes depend on the number of iterations or the predefined percentage

error limit. After the predefined number of iteration or the maximum relative

percentage error for the unknown variables was satisfied, the TPNS simulation gave

136

the value of each of the unknown variables, compression ratio, and power

consumption for the system. The results were displayed on the basis of iterations.

The rate of convergence to the final solution depends on the initial estimation and

usually high at the beginning of the iterations. Based on the pipe data shown in

Table 4.7, source pressure of 3500kPa and an end pressure requirement of 4000kPa,

compressor speed of 7600 rpm, the results of the unknown nodal pressure and flow

variables were obtained for the first ten iterations. At the end of the 10th iteration, the

maximum relative percentage error was obtained to be 8.66347E-17. The results

from the TPNS simulation for the unknown nodal pressures and flow variables are

shown in Table 4.8. The corresponding compression ratio and power consumption

for the system were obtained to be 1.4951 and 10.9378 MW, respectively.

Table 4.8 Results of nodal pressures and flow variables after 10 iterations

Node Pressure[kPa] Main flow m3/hr Branch flow m3/hr

0 3500.00 Q1 770480.0 QC1 135404.0

1 2779.23 Q2 142038.0 QC2 135404.0

2 4155.23 Q3 357634.0 QC3 59865.3

3 4104.47 Q4 270808.0 QC4 64933.0

4 4066.28 Q5 124798.0 QC5 134465.0

5 4006.89 Q6 232836.0 QC6 69509.2

6 4031.19 Q7 98371.1 QC7 76459.2

7 4023.17 Q8 28861.9 QC8 41252.3

8 4022.43 Q9 94440.5 QC9 1726.45

9 4002.95 Q10 53188.2 - -

10 4002.74 Q11 51461.8 - -

The convergences of the pressure variables at node 1 and 2 and the main flow

variable 1Q were studied under various initial estimations.

Figure 4.19 and Figure 4.20 show the convergence of nodal pressures to the final

solutions for the first ten iterations at node 1 and 2 of the pipeline network system

137

shown in Figure 3.22 based on the initial estimations of 2000, 4000 and 10000kPa.

The convergence of the remaining nodal pressures can also be plotted following the

same procedures as that of the nodal pressures at node 1 and 2.


138


From the study of the convergence of the nodal pressures at node 1 and 2, it was

observed that the TPNS simulation model could provide solutions to pressure

variables with a wide range of initial estimations. From the simulation experiments,

the initial estimation near to the end pressure requirement was obtained to be a good

initial estimation and resulted solutions in all the tests conducted. The convergence

of the main flow parameter 1Q for the initial estimations of 2000, 4000 and

10000m3/hr is shown in Figure 4.21.

139


The convergence of the flow variables through the main pipes and the lateral

pipes were also plotted. Figure 4.22 shows the convergence of the remaining main

flow variables for the first ten iterations. As indicated in the figure, almost all the

main flow variables solutions were obtained starting from the third iteration.

140

Figure 4.22 Convergence of the main flow variables

iii) Application of TPNS Simulation for Compressor Performance Analysis

Further simulation study was conducted in order to apply the TPNS simulation

model for performance analysis of the compressor used for the pipeline network

system shown in Figure 3.22. For the performance analysis of the compressor,

suction pressure (P1), discharge pressure (P2) and the flow rate through the

compressor (Q1) were considered. The performance of the compressor was studied

for source pressure of 3500kPa to meet pressure requirements of various demand

pressures ranging from 4000 to 5000kPa. The analysis was performed with

compressor speeds of 5000, 5500, 6000 and 6500rpm. Note that the analysis can also

be made at any speed of the compressor as long as the speed is within the working

limit of the compressor. The results of the application of the Newton-Raphson based

TPNS simulation for performance analysis was reported in [92].

141

The variation of the discharge pressure (P2) of the compressor with flow rate (Q1)

is shown in Figure 4.23. As it is observed from the figure, for a constant speed

operation of the compressor, an increased in discharge pressure gave rise to a

decrease in flow capacity of the system and vice versa. This showed that the

characteristic map generated using the TPNS simulation model is similar to the

characteristics maps of the compressors described in [60, 93]. As a result, the

simulation model could be used to analyze the performance of the compressors.

Figure 4.23 Discharge pressure variations with flow for various speeds

The variation of CR with flow and speed is shown in Figure 4.24. As it is seen

from the figure, higher CR for the system was achieved at lower flow capacities.

This is because, based on equation (3.76), power consumption of the system is a

function of CR, flow rate and other properties of the gas. From this equation, it is

observed that lower flow rate results higher CR values. For a constant flow

operation, an increase in speed of the compressor increased the CR of the system.

142

The results in the figure also showed the characteristic map of the compressor based

on CR which is similar in trend with that of the characteristic map plotted in [60].

Figure 4.24 Compression ratio variations with flow for various speeds

Power consumption variation with flow rate through the compressor based on

various speed is shown in Figure 4.25. It is observed that an increase in flow rate

increased the power consumption. For a constant flow operation, an increase in speed

of the compressor increased the power consumption of the system. As it is given in

[23], power consumption by the system is function of the flow rate, CR and

properties of the gas. When the speed of the compressor increased at constant flow

operation, it is shown in Figure 4.24 that the CR of the compressor is also increased.

Hence, the power consumption of the system increased due to an increment in CR of

the system.

143

Figure 4.25 Variation of power consumption with flow for various speed

4.3.5 The Case of Divergent in TPNS Simulation Model

The Newton-Raphson based TPNS simulation model plays significant role in

achieving the required pressure, flow, and temperature variables usually within four

to six iterations depending on the error tolerance limit. However, the Newton-

Raphson solution scheme has limitations depending on the initial estimations as

reported in [81]. Initial estimations which are too far from the final solution usually

result in divergence. In order to guide for successful convergence, several attempts

were conducted on TPNS simulation model and an initial estimations which gives

solution for the variables were identified. Initial estimations near to the demand

pressure requirements resulted in convergence for the attempts made on TPNS

simulation model. Sample of the divergent case for branched TPNS is shown in

Appendix D.

144

4.4 Enhanced Newton-Raphson based TPNS Simulation Model

The results and discussions in this section are based on the enhanced Newton-

Raphson based TPNS simulation model presented in section 3.5 on page 88. The

model was applied for analyzing gunbarrel and branched pipeline network

configurations considering two-phase gas-liquid flow analysis, single phase with

corrosion effect and single phase with temperature variations.

4.4.1 Two-phase Gas-liquid Flow Analysis

The analyses conducted in this section for the gunbarrel and branched network

configurations were based on the methodology discussed in section 3.5.1 on page 88.

For the analysis of the networks, it was assumed that the liquid holdup is

005.0== LLH λ which is very common in transmission network system [61]. As a

result, the density of the mixture and the viscosity of the mixture were determined to

be 3/7425.5 mkgm =ρ and mskgEm /0599.2 −=μ , respectively.


The results and discussions presented in this section are based on the gunbarrel

TPNS network configurations shown in Figure 3.19 on page 77. The analysis is

based on the pipe data shown in Table 3.14 and the nodal pressure requirements data

shown in Table 3.13. The network was analyzed based on the enhanced Newton-

Raphson based TPNS simulation model considering effect of two-phase flow

analysis. The same compressors as in section 4.3.1 with speed of 8000 rpm were

used for the analysis. The results for the unknown pressure variables and flow

variable were obtained with less than ten iterations. The convergences of nodal

pressures and flow variable for the network system were studied. Based on the

results obtained from the iterations, the convergence of the pressure variables and

flow variable are as shown in Figure 4.26 and Figure 4.27, respectively.

145

Figure 4.26 Convergence of pressure variables (Case 1C)

Figure 4.27 Convergence of flow variable (Case 1C)

146


The results and discussions presented in this section are based on the branched

TPNS network configurations shown in Figure 3.20 on page 79. The network was

analyzed using the enhanced Newton-Raphson based TPNS simulation model

considering the effect of two-phase flow. The analyses regarding the network were

based on the pipe data shown in Table 3.17. The same compressor specifications

with speed 7800rpm was used as in 4.3.2. The pressure at node 0 was 3500kPa and

the demand pressure from the customer was assumed to be 4000kPa.

The results for the unknown pressure variables and flow variables were obtained

with less than ten iterations with relative percentage error of 8.16639E-17. Figure

4.28 shows the convergence of nodal pressures to the final pressure solution for the

first ten iterations with an initial estimation of 2000kPa for all pressure variables and

2000m3/hr for all flow variables. The simulation model was also tested by giving

wide range of initial estimations and convergence was achieved in most of the trials.

An initial estimations near to the demand pressure resulted in convergences in all the

tests conducted on the pipeline network system. Most of the pressure parameters

converged to their final solution after the third iterations.

147


The flow through main pipe joining node 5 and node 6 is less compared to the

other main pipe flows. As a result, the pressure drop from node 5 to node 6 is small

compared to the other main pipes. Hence, the convergence of the nodal pressure at

node 5 and node 6 coincides nearly to the same line. The result from the simulation

model showed that the pressure at end of the tenth iteration at node 5 and 6 were

4013.48 kPa and 4008.43kPa, respectively. The convergence of the corresponding

flow variables are shown in Figure 4.29 and Figure 4.30.

148

Figure 4.29 Convergence of main flow variables (Case 2C)

Figure 4.30 Convergence of lateral flow variables (Case 2C)

149

The effect of the variation of the liquid holdup on nodal pressures and flow

through pipes were also studied. Figure 4.31 shows the effect of liquid holdup on

nodal pressures. Since the customer requirements have to be satisfied in terms of

pressure, the nodal pressures remain nearly the same for various liquid holdups.

However, slight increment in pressure drop was observed when the liquid holdup

increased.

Figure 4.31 Effect of liquid holdup on nodal pressures

Liquid holdup had a significant effect on flow through pipes. An increase in

holdup decreased the flow through pipes as shown in Figure 4.32. This is because,

when the liquid holdup increased, flow resistance increased to result in a decrease in

flow of the mixture through pipes.

150

Figure 4.32 Effect of liquid holdup on flow through main pipes

4.4.2 The Effect of Internal Corrosion

The analyses conducted in this section for the gunbarrel and branched network

configurations were based on the methodology discussed in section 3.5.2 on page 96.


The gunbarrel pipeline network shown in Figure 3.19 on page 77 was also

analyzed with the aid of the enhanced Newton-Raphson based TPNS simulation

model by considering the effect of corrosion. The same specifications of compressors

were used as in section 4.3.1. The age of the pipes could vary to give the

corresponding friction coefficient which results different pressure drop. The results

of the unknown nodal pressures and flow variable were obtained for the first ten

iterations for ten years old pipes. Based on these results, the convergence of nodal

151

pressures and flow variables for the network system are shown in Figure 4.33 and

Figure 4.34, respectively.

The comparison of the convergence graphs obtained by neglecting the effect of

corrosion in Figure 4.7 is identical to the convergence graph obtained considering the

effect of corrosion shown in Figure 4.33. However, for the case of corrosion large

shootings are observed on each graph from the initial estimation. This is resulted

from the behavior of the convergence in Newton-Raphson solution technique where

the graphs always tend towards the solution depending on the initial estimations.

Initial estimations less than the final solutions are always resulted the convergence

graphs to shoot upward before going stable at the solution. On the other hand, an

initial estimation which are greater than the final solution resulted the convergence

graphs to shoot downward before coming to stable at the final solutions.

Figure 4.33 Convergence of pressure variables with corrosion (Case 3C)

152

Figure 4.34 Convergence of flow variable with corrosion (Case 3C)

The results shown in Table 4.1 on page 115 were compared with the results

obtained by considering the effect of corrosion for various ages of the pipe. The

pressures at node 0 and 5 were constant throughout the various ages. This is because

the pressures at these nodes are the given conditions. The speed of the compressor is

assumed to be 8000 rpm for both cases. For the case of neglecting the effect of

corrosion, only the natural frictional coefficient of the pipe was considered which is

constant throughout the service of the pipes. However, for the case of pipes where

carrions is taken into account, the friction factor varied with service life of the pipe

based on the relationships developed in section 3.5.2 on page 96. The frictional

coefficient had an effect on the pressure drop, flow capacity, compression ratio and

power consumption.

Table 4.9 shows the comparison of the pressure variables at different nodes for

various ages of the pipes. As indicated in the table, higher pressure drops were

observed as the ages of the pipe increased. For instance, for pipe joining node 0 and

153

1, the pressure drop for the 20 years old pipe was higher than the pressure drop for

10 years old.

Table 4.9 Comparison of nodal pressures for different pipe ages

Nodes

Pressure [kPa] for various ages

New pipe 10 years

old pipe

20 years

old pipe

0 3000.00 3000.00 3000.00

1 2472.77 2460.22 2447.55

2 3505.01 3504.79 3504.51

3 3065.90 3055.53 3045.02

4 4345.73 4352.85 4359.99

5 4000.00 4000.00 4000.00

The flow capacity of the system was also studied for various ages of the pipes.

The variation of the flow capacity of the pipe with age of the pipe is shown in Figure

4.35. From the figure, it is observed that the capacity of the pipe is reduced as the

ages of the pipe increased. This is because, as the age of the pipes increase the

roughness of the pipe increases due to corrosion and accumulation of various

elements. As a result, the friction factors for the pipes increased which increase the

pipe flow resistance. An increase in flow resistance of the pipe decreases the flow

capacity of the system.

154

Figure 4.35 Variation of flow with age of the pipe

The variation of CR with ages of the pipe is shown in Figure 4.36. CR increased

as the age of the pipe increased. This is due to the fact that, an increase in ages of the

pipe could result in an increase in roughness of the pipe which leads to increase in

friction factor. As a result, the pressure drop increased which cause higher CR.

155

Figure 4.36 Variation of compression ratio with age of the pipe

The variation of the power consumption by the system with age of the pipe is

shown in Figure 4.37. As it is observed from the figure, for constant speed operation,

power consumption decreases with an increase in age of the pipes. As shown in

equation (3.62), power consumption of the system is mainly function of the flow

rate, CR and properties of the gas. From Figure 4.35, it is observed that an increase

in age of the pipe caused a decrease in flow capacity of the system. Moreover, it is

observed in Figure 4.36, CR increased with age of the pipes. Therefore, the trend of

power consumption with age of the pipes depends on either flow rate or CR. From

Figure 4.37, it is observed that the trend of power consumption followed the same

trend as that of Q1. This showed that the effect of flow variation was more dominant

than the effect of CR variations with age of the pipes on power consumption.

156

Figure 4.37 Variation of the power consumption with age of the pipes


The branched pipeline network shown in Figure 3.20 on page 79 was also

analyzed with the aid of the enhanced Newton-Raphson based TPNS simulation

model by considering the effect of corrosion. The same specifications of compressors

were used as in section 4.3.2. The age of the pipes could vary to give the

corresponding friction coefficient. The convergences of the pressure and flow

variables were also studied based on ten years old pipes. The convergence of some of

nodal pressure variables is shown in Figure 4.38. It is observed from the graph that

the solutions for the nodal pressures were obtained after the third iterations. The

maximum error at the end of the 10th iterations was 1.92x10-15. Figure 4.39 shows the

convergence of flow variables through main pipes. As it is observed from the graph,

the TPNS simulation nearly arrived to the final solution at the 4th iteration. At the end

of the 4th iteraion, the maximum relative error was 0.045935.

157


Figure 4.39 Convergence of flow through main pipes (Case 4C)

158

The convergence of flow variables through the branch pipes is shown in Figure

4.40. Similar to the convergence of the nodal pressures and flow variables through

main pipes, the TPNS simulation converged almost to the final solution at the end of

the 4th iteration for the flow variables through lateral pipes.

Figure 4.40 Convergence of flow through lateral pipes (Case 4C)

The results obtained in section 4.3.2 were compared with the results obtained by

considering the effect of corrosion for various ages of the pipe. Three groups of

pipes i.e. pipes with ages of 0, 10 and 20 years old were considered for comparison.

Pipes with 0 years old are considered to be as new pipe. For this pipe, only the

natural roughness of the pipe was considered during the analysis. Figure 4.41 shows

the values of the pressure variables for different ages of the pipe. Since the customer

requirement should have to be satisfied in terms of pressure in all groups of pipes,

the nodal pressures remain nearly the same for different ages of pipes. However,

slight increment in pressure drop is observed when the ages of the pipes increased.

159

Figure 4.41 Nodal pressures for different ages of pipes

The variation of flow variables with ages of pipes is shown in Figure 4.42. It is

observed that corrosion had a significant effect on flow capacity of the pipes. The

results of the simulation analysis showed that a decrease in flow capacity of 2.16%

and 4.35% were observed for the 10 and 20 years old pipes, respectively.

160

Figure 4.42 Flow through pipes for different ages of pipes

Table 4.10 shows the results of the simulation study conducted on pipe joining

node 0 and 1 of the pipeline network system. The friction factor, pipe flow resistance

and the flow through the pipe were studied under three age groups. As the age of the

pipes increased, the friction factor and pipe flow resistance increased. Based on flow

equations, for fixed pressure at the start and end node of the pipe, an increase in flow

resistance of the pipe reduced the flow capacity.

Table 4.10 Variation of flow with ages of pipes

Years in service

Friction Pipe flow resistance Flow[m3/hr]

New pipe 0.007003 9.01E-06 466683.00

10 0.007407 9.53E-06 456596.00

20 0.007847 1.01E-05 446371.00

161

4.4.3 Non-isothermal Flow Analysis

The analyses conducted in this section were based on the methodology discussed

in section 3.5.3 on page 101. The gunbarrel pipeline network shown in Figure 3.19

on page 77 was also analyzed with the aid of the enhanced Newton-Raphson based

TPNS simulation model for non-isothermal conditions. The same compressors data

as in section 4.3.1 was used for the analysis. The source temperature was assumed to

be 310 K. The results for the unknown pressure, flow and temperature variables were

obtained with less than ten iterations. The unknown temperature variables for the

first ten iterations are shown in Table 4.11. Note that temperature is measured in K.

Table 4.11 Results of temperature variables for the first 10 iterations

Iteration T1 T2 T3 T4 T5 Max. error 0 4000.00 4000.00 4000.00 4000.00 4000.00

1 360.576 466.115 97337.9 97443.5 98018.4 0.998674

2 309.982 237.188 237.263 -16406.9 -16538.2 0.602838

3 309.972 827.606 822.405 -40046.9 -39564.5 0.504891

4 309.961 581.863 575.621 13105.1 13051.8 0.310895

5 309.948 430.935 426.949 -2749.02 -2753.84 0.351659

6 309.939 362.365 360.53 926.084 917.115 0.107755

7 309.938 358.562 357 408.03 404.835 0.006933

8 309.938 358.469 356.91 412.797 409.561 2.79E-05

9 309.938 358.468 356.91 412.795 409.559 3.02E-09

10 309.938 358.468 356.91 412.795 409.559 1.52E-14

The convergence of the pressure variables and flow variables are shown in

Figure 4.43 and Figure 4.44, respectively.

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Figure 4.43 Convergence of pressures for non-isothermal conditions

Figure 4.44 Convergence of flow variables under non-isothermal conditions

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Based on the results obtained in Table 4.11, the convergence of some of the

nodal temperature variables for the network system is shown in Figure 4.45. The

convergences of the remaining nodal temperatures follow the same trends as T1. It

required a minimum of 6 iterations for the error to be stable. This is mainly due to

the initial estimations which were far from the final solution of temperature

variables.

Figure 4.45 Convergence of some of the temperature variables

4.5 Verification and Validation of the TPNS Simulation Model

The developed TPNS simulation model provided the required operational

variables of the pipeline network for various configurations. The results from the

simulation model needs simulation verification and validation for accuracy and

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reliability. According to [94], model validation is usually defined to mean

“substantiation that a computerized model within its domain of applicability

possesses a satisfactory range of accuracy consistent with the intended application of

the model”. Model verification is often defined as “ensuring that the computer

program of the computerized model and its implementation are correct”. These

definitions of validation and verification are adopted in this thesis.

Since simulation models are increasingly being used in problem solving and in

decision making in various areas of interest, the verification and validation

techniques also varies depending on the types and nature of the model. The different

techniques on simulation model verification and validation are presented in

[95]-[97].

The availability of real system data gives more rooms to implement the various

verification and validation techniques on the simulation model. As presented in [98],

there are three situations concerning the data availability which includes, the case of

no data, the case of only output data, and the case of both input and output data

availability. Various model validations and verifications were suggested based on the

nature of the availability of the data.

The TPNS simulation model case is different and could not be categorized as the

situations reported in [98]. This is because, the TPNS simulation model is developed

based on the actual performance of the compressors, gas properties, pipe

information, and source and exit pressure requirements. As a result, some of the

input parameters are known. Even though the output data from the real system are

available, some data were considered to be confidential by the company. Hence, it

would be difficult to categorize the situations of the TPNS simulation model as one

of the classifications discussed based on the data availability.

Simulation model validation and verification are not considered as an end

process. They are integrated with the modeling process. Sargent [95] presented how

the simulation verification and validation process are integrated with the modeling

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process. Sargent framework for modeling process is used as a basis in this research

work. Figure 4.46 shows the simplified framework for simulation modeling process.

By referring to Figure 4.46, the problem entity is the system (real or proposed),

idea, situation, policy, or phenomena to be modeled; the conceptual model is the

mathematical/logical/verbal representation (mimic) of the problem entity developed

for a particular study; and the computerized model is the conceptual model

implemented on a computer. The conceptual model is developed through an analysis

and modeling phase, the computerized model is developed through a computer

programming and implementation phase, and inferences about the problem entity are

obtained by conducting computer experiments on the computerized model in the

experimentation phase [95].

Figure 4.46 Simplified version of the modeling process [95]

4.5.1 TPNS Conceptual Model Validation and Verification

Conceptual model validation is one of the fundamental steps that were included

during the development of the TPNS simulation model. This phase of the model

development process includes the determination of the theories and assumptions

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underlying the conceptual model are correct and that the model representation of the

problem entity is reasonable for the intended purpose of the model.

The various assumptions, theories and the relationships between parameters were

studied carefully in order to insure the conceptual model validity. Some of the

procedures conducted consist of thorough discussion with expert on the basic

principles for model development and studying the basic relationships between the

flow, pressure, temperature and performance measurements. Furthermore the whole

model was divided into sub-models and studied if appropriate structure, logic, and

mathematical relationships have been used. For instance, the pipe flow equations

which is one of the governing simulation equations was divided into various sub

groups in order to make sure that the mathematical relationships and basic principles

were properly implemented.

Proper examination of the flowchart of the model and plotting of the basic

relationships of the parameter were also conducted on the TPNS conceptual

simulation model. The general flow equation was plotted based on pressure versus

length of pipe for various flows in order to study the pressure drop profile. A

comparison with the known flow equations is also performed. The compressor

equation was derived by using some of the compressor data. The remaining data was

used to validate the compressor equations as shown in Figure 3.9.

The model verification procedures ensure that the computer programming and

implementation of the conceptual model are correct. There are various techniques

suggested in [95] for simulation model verification. Two types of tests were

conducted on the developed TPNS simulation model in order to ensure that the

computer program is properly implemented. The first test was done based on static

testing principles. This test consists of analyzing the computer program to determine

if it is correct by using correctness proofs, structured walk-through and examining

the structure properties of the program. The second test was conducted based on

dynamic testing principles. In dynamic testing, the computerized TPNS simulation

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model was executed under different conditions like single phase flow analysis, two-

phase flow analysis, and single phase with no corrosion effect. Furthermore, the

TPNS model was tested for gunbarrel, branched and looped pipeline networks which

are the most commonly used configurations. The resulting values from the dynamic

analysis of the TPNS simulation model have shown that the computer program and

its implementations were correct. The techniques commonly used in dynamic testing

suggested in [94] are traces, investigations of input-output relations using different

validation techniques, internal consistency checks, and reprogramming critical

components to determine if the same results are obtained.

4.5.2 Operational Validation

One of the techniques applied for validation of TPNS simulation is the

operational validation techniques. This technique ensures whether the TPNS

simulation model’s output behavior has the required accuracy for the application of

the model to real system. This is where much of the validation testing and evaluation

take place. There are various simulation validation techniques suggested in

literatures [95], [97] and [99]. The type of validation techniques applied for the

simulation model depends on the nature of the simulation model and the availability

of data. The following sections discuss the detail of the operational validation

techniques applied on the TPNS simulation model.

4.5.2.1 Comparison to Other Models

One of the techniques used in order to validate the TPNS simulation model was

comparison with the previous models. The lack of data from literatures with the same

problem instances make difficult to validate the simulation model based on model

comparison. However, there were various types of pipeline networks analyzed for

determination of optimal parameters or determination of the nodal pressure and pipe

flow parameters for the given conditions. Two cases were taken from the available

literatures in order to validate the results of the developed TPNS simulation model.

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As discussed in sections 4.3.1 and 4.3.3, the results of the TPNS simulation

model were compared with two other models based on various pipeline network

configurations. Even though complete data regarding the type of compressors used

were not mentioned in the cases considered, the TPNS simulation was able to

provide solutions in both instances. In the first case, the TPNS simulation model was

compared to an exhaustive optimization technique based on gunbarrel pipeline

networks system. As discussed in detail in section 4.3.1, TPNS simulation yielded

close solutions of nodal pressures and flow variables to the exhaustive technique.

The comparison based on looped pipeline network discussed in section 4.3.3 also

showed that the TPNS simulation model was able to provide solutions to nodal

pressures and flow variables which were close to previous model.

4.5.2.2 Validation based on Operational Graphics

As presented in [94], this type of validation techniques consists of showing

graphically the values of performance measures as the model moves through time or

subjected to variations. The dynamical behavior of performance indicators are

visually displayed as the simulation model runs with variations to ensure they behave

correctly.

The operational validation technique was demonstrated based on the pipeline

network shown in Figure 3.22. The performance of the compressor for the pipeline

network was presented in section 4.3.4. The performance analyses of the compressor

include the flow capacity, power consumption and compression ratio. The

performance characteristics maps generated using the developed simulation model

was similar to the one available in the literatures.

4.5.2.3 Model Parameter Variability - Sensitivity Analysis

One of the techniques used in order to validate the simulation model was

sensitivity analysis. This technique consists of changing the values of the input and

internal parameters of the model to determine the effect upon the model’s behavior

169

and its output. As presented in [100]-[102], performing sensitivity analysis on

simulation model is one of the key elements in studying the effect of the input

parameters variations on the output data.

Sensitivity analysis was conducted on the TPNS simulation model by varying the

input parameters like compressor speed, number of compressors working within the

compressor station and age of the pipe. The effect of the variation of these

parameters on the TPNS output parameters and model behavior was studied.

Sensitivity analysis was conducted on the part of the existing pipeline network

(Figure 3.22) which was used to demonstrate the application of the TPNS simulation

model for real system. The pipeline data shown in Table 4.7 was used for the

analysis. It was assumed that the demand pressure at various customer stations to be

4000kPa. The analysis was done based on single compressor data obtained from

[59]. The details of the sensitivity analysis conducted for the TPNS by varying the

input parameters are discussed as follows.

i) Variation in number of compressors working within the station

The variations of the number of compressors on the performance of the system

were studied based on constant speed operations of the compressors. Compressors

speed of 7500, 8000, 8250, and 8500rpm were taken as constant speed operations.

The maximum compression ratio of the system was limited to 1.5. Furthermore, the

source pressure for the pipeline network was 3000kPa and the age of the pipes was

assumed to be ten years.

Figure 4.47 shows the variation of main flow variable of the pipeline network

system (Q1) with number of compressors working within the station for different

speed operation of the compressors. When the number of compressors working

within the station is increased, it is observed that the flow capacity of the system

increased. As it is seen from equation (3.10), an increase in number of compressor

improves the flow capacity of the system. There will be more compressors to work

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and satisfy the requirements. However, significant increment in flow rate capacity of

the system was observed when the number of compressors working within the station

increased to maximum of 4. The effect of the addition of a compressor to the existing

compressors became less as the number of compressors increased from 5. Therefore,

flow rate capacity of the system is more sensitive to increase in number of

compressors only at lower number of compressors.

Figure 4.47 Variations of flow rate with number of compressors

Figure 4.48 shows the compression ratios when the number of compressors

working within compressor station varies for different speed operations. An increase

in number of compressors working within the compressor station caused an increase

in compression ratio of the system for various speed operations. As seen from Figure

4.47, an increase in number of compressors enhanced the flow capacity Q1. Hence,

based on equation (3.1), an increase in flow capacity leads to higher pressure drop

which increase the CR. Significant changes in CR was observed for all speed

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operations when the number of compressor changed from 1 to 2. Even though an

increase in number of compressors increased the compression ratio of the system,

their effect reduced as the number of compressors increased.

Figure 4.48 Variation of compression ratio with number of compressors

The magnitude of the variation of CR for various constant speed operations

became more sensitive to change in number of compressors. For instance, when the

number of compressor working within the system was 1, CR for compressor speed of

7500 rpm and 8500 rpm was 1.21459 and 1.24845, respectively. However, an

increase in number of compressors working within the system to 5 gave CR of

1.37902 and 1.50166 for compressor speed of 7500rpm and 8500rpm, respectively. It

is observed from Figure 4.48 that, compression ratio is more sensitive at lower

number of compressors and the deviations among the constant speed operations

increased with increase in number of compressors.

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Figure 4.49 shows the variation of the power consumption with number of

compressors at various speed operations. In the case of the power consumptions with

number of compressors working, almost the same scenarios were observed as that of

the compression ratio variations.

Figure 4.49 Variation of power consumption with number of compressors

At lower number of compressors working within the station, power consumption

was more sensitive compared to high number of compressors working within the

system. The deviations in power consumption at various speed operations were

increased with increase in number of compressors working within the system.

ii) Variation in speed of the compressors working within the station

For performing the sensitivity analysis based on the variation in speed of the

compressors, the number of compressors working within the station was assumed to

be two. The source pressure for the pipeline network was 3500 kPa and the required

173

demand pressures were assumed to be 4000, 4250 and 4500kPa. The age of the pipes

was assumed to be ten years. The variation of flow capacity of the system,

compression ratio and power consumption with speed were studied and presented in

section 3.4.4.

iii) Variation in source pressure

For performing the sensitivity analysis based on the variation in source pressure,

three constant speeds operations of compressor were identified, i.e. 8000rpm,

8500rpm and 8750 rpm. The simulation can be performed any arbitrary speed

provided that the speed selected are within the working limit of the compressor. The

demand pressure for the system was 4500kPa. The age of the pipes was assumed to

be ten years. The number of compressors working within the compressor station was

two. The working range of source pressures for each speed is bounded by the

maximum CR and speed of the compressor. For instance, at source pressure of

3200kPa and compressor speed of 8750 rpm, the CR was 1.51074. Any source

pressures less than 3200kPa could result higher CR which is beyond the working

limit.

Figure 4.50 shows the variation of main flow rate (Q1) with variation in source

pressure (P0). For fixed compressor speed and demand pressure requirement, an

increase in source pressure increased Q1 and vice versa. For high speed operation of

the compressors, high increment in the main flow was observed compared to low

speed operations.

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Figure 4.50 Variation of main flow rate with source pressure

As it is shown in Figure 4.51, an increase in source pressure decreased the

compression ratio of the system for constant speed operation and demand

requirement. In real system operation of the compressors, in order to meet fixed

demand pressure requirement at constant speed, an increase in source pressure makes

the system to move downward on the performance map of the compressor following

the trace of constant speed line. This causes an increase in flow rate capacity and a

decrease in compression ratio of the system. Thus, both the main flow Q1 and the

compression ratio of the system are highly sensitive to change in source pressure.

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Figure 4.51 Variation of compression ratio with source pressure

Figure 4.52 shows the variation of power consumption of the pipeline network

system for various source pressures. An increase in source pressure increased the

power consumption the system. Power consumption by the system is mainly

dependent on the flow rate capacity of the system, compression ratio and properties

of the gas as presented in [23]. As shown in Figure 4.50 and Figure 4.51, an

increased in source pressure increased the flow capacity and decreased the

compression ratio of the system. Therefore, the overall effect of change in source

pressure is either to increase or decrease the power consumption of the system

depending on which factor is dominating. From Figure 4.52, it is observed that an

increase in source pressure increased the power consumption of the system which

indicates that the flow capacity of the system is the dominating factor over the

compression ratio.

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Figure 4.52 Variation of power consumption with source pressure

iv) Variation in the age of the pipes

The variations of the flow capacity, compression ratio and power consumption of

the system with the age of the pipes were also studied. For performing the sensitivity

analysis based on the variation in the age of the pipes, the speed of the compressor,

source pressure, and the number of compressors working within the system were

assumed to be constant. The details of the effect of the age of the pipes on the

performance measures of the system were presented in section 4.4.2.

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4.6 Summary

The simulation studies based on successive substitution and Newton-Raphson

method showed that the TPNS simulation model produced solutions to nodal

pressures and flow variables for various pipeline network configurations. Successive

substitution scheme was limited to simple pipeline network configurations as the

method is highly dependent on information flow diagram which is a difficult task to

produce when the number of unknowns increased. The applications of successive

substitution scheme on two different pipeline network configurations showed that, it

took many iterations to arrive at specified tolerance.

The simulation experiments were conducted on TPNS simulation model based on

Newton-Raphson solution scheme for the three most commonly found

configurations, i.e. gunbarrel, branched and looped pipeline network system. It took

less than ten iterations to arrive at reasonable error tolerance limit. Usually, it took a

maximum of 4 iterations for the TPNS simulation model for the error to arrive at the

stable solutions.

As discussed in sections 4.3.1 and 4.3.3, the results of the Newton-Raphson

based TPNS simulation model were compared with two other models based on

various pipeline network configurations. Even though complete data regarding the

type of compressors used did not mentioned in the cases considered, the Newton-

Raphson based TPNS simulation was able to provide solutions in both instances. In

the first case, the simulation model was compared to an exhaustive optimization

technique based on gunbarrel pipeline networks system. As discussed in detail in

section 4.3.1, simulation model yielded close solutions of nodal pressures and flow

variables to the exhaustive technique. The comparison of TPNS simulation model

based on looped pipeline network configuration was discussed in section 4.3.3. The

model produced solutions to nodal pressures and flow variables which were close to

previous model.

178

The application of the Newton-Raphson based TPNS simulation model for real

pipeline network system was also conducted based on existing pipeline network

system in sections 3.4.4. Three modules of TPNS simulation model which includes

input parameter analysis, function evaluation and network evaluation module were

evaluated using the data taken from the real system. Analyses of the performance of

compressor for existing pipeline network system which includes discharge pressure,

compression ratio and power consumption were also conducted using the developed

Newton-Raphson based TPNS simulation model. The performance characteristics

maps generated by the simulation model showed the variation of discharge pressure,

compression ratio, and power consumption with flow rate as similar to the one

available in the literatures.

The enhanced Newton-Raphson based TPNS simulation model was also applied

for analyzing branched and gunbarrel network configurations considering two-phase

gas-liquid flow analysis, single phase with corrosion effect and single phase with

temperature variations. In all cases, the results for the required nodal pressures,

temperature and flow variables were achieved with less than ten iterations to the

reasonable relative percentage errors. For instance, for two phase flow analysis

discussed in section 4.4.1 on page 144 for case 2C, the relative percentage error at

the end of the 10th iteration was obtained to be 8.16639x10-17.

As discussed in section 4.5, the TPNS simulation model was verified and

validated using the various techniques. The model verification and validation was

integrated with the TPNS modeling process. Conceptual model validation and

verification and operational validations were conducted on the TPNS simulation

model based on the available data and simulation experiment.

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CHAPTER 5

CONCLUSIONS AND RECOMMENDATIONS

5.1 Conclusions

In order to evaluate the performance of natural gas pipeline transmission network

systems with non-pipe elements, a TPNS simulation model was developed. The

developed TPNS simulation model was able to provide solutions to nodal pressure

and flow variables for the given pipeline network systems. The performances of the

system were analyzed based on the variables obtained. The TPNS simulation model

was also enhanced to analyze pipeline network systems under two-phase gas-liquid

flow, variable ages of pipes, and variable temperature.

The developed TPNS simulation model incorporated detailed parameters of the

compressors into the governing simulation equations as discussed in section 3.2.3 on

page 52. Compressor stations should not be considered as a black box or represented

with few parameters during simulation. Speed of the compressor, flow rate, suction

pressure and discharge pressure were the critical elements which affected the

performance of the system as presented in section 3.4.4.

The TPNS simulation model proposed in this thesis based on the iterative

successive substitution and Newton-Raphson algorithm were tested for different

network configurations. The implementation of TPNS simulation model based on

successive substitution scheme discussed in section 4.2 showed that the model is

more suitable to simple network configurations. Simulation experiments were

conducted on TPNS simulation model based on Newton-Raphson solution scheme

for the three most commonly found configurations, i.e. gunbarrel, branched and

180

looped pipeline network system as presented in section 4.3. In all the investigations,

the solutions to the unknown variables were obtained with a wide range of initial

estimations. A maximum of 10 iterations were required to get solutions to nodal

pressure and flow variables with relative percentage errors less than 10-11.

The results of the Newton-Raphson based TPNS simulation model were

compared with two other models based on various pipeline network configurations.

In the first case, the simulation model was compared to an exhaustive optimization

technique based on gunbarrel pipeline networks system. The model yielded close

solutions of nodal pressures with less than 1.8% absolute parentage errors. The

comparison based on looped pipeline network also showed that, the Newton-

Raphson based TPNS simulation model was able to provide solutions to nodal

pressures and flow variables with mean absolute error of 5.10 % between the two

methods.

The application of the Newton-Raphson based TPNS simulation model for real

pipeline network system was also implemented based on the existing pipeline

network system as discussed in section 3.4.4. Three modules of TPNS simulation

model which includes input parameter analysis, function evaluation and network

evaluation module were evaluated using the data taken from the real system.

Analyses of the performance of compressor for existing pipeline network system

which included discharge pressure, compression ratio and power consumption were

also conducted using the developed TPNS simulation model. The performance

characteristics maps generated by the developed TPNS simulation model show the

variation of discharge pressure, compression ratio, and power consumption with flow

rate similar to the one available in the literatures.

Simulation analysis was conducted using the enhanced Newton-Raphson based

TPNS simulation model to investigate the effect of the ages of pipes on the

performance of the system based on gunbarrel and branched network configuration

as discussed in section 4.4.2. Pressure drop and flow rate of the gas were affected as

181

the age of the pipe increases. The comparison of the performances of the three

groups of pipes based on new, 10 years old, and 20 years old having branched

network configurations showed that a decrease in flow capacity of 2.16% and 4.35%

was observed for the 10 and 20 years old pipes, respectively. Since the customer

requirement have to be satisfied in terms of pressure in all groups of pipes, the nodal

pressures remain nearly the same for different ages of pipes as shown in Figure 4.41.

For instance the nodal pressure at node 1 increased by only 0.119% when the years

in service of the pipe increased to 10 years. The nodal pressure also increases by

0.239% when the years in service increased to 20 years.

The proposed enhanced Newton-Raphson based TPNS simulation model consists

of module to analyze pipeline network systems with two-phases by incorporating

modified homogeneous flow equation into the governing simulation equations. As

presented in section 4.4.1 on page 144 , the simulation analysis conducted on

branched TPNS configuration revealed that liquid holdup has a significant effect on

flow capacity of the system. For instance, for the main flow rate Q1, the flow

capacity reduced by 54.56% when the liquid holdup increased from 0.0001 to 0.005.

The corresponding nodal pressure at node 1 increased by 0.73%. Even though small

amount of liquid holdup, usually less than 0.005 [24], is found in transmission

pipeline network systems, the existence of the liquid had effect on pressure drop and

flow capacity of the system.

5.2 Contributions of the Research

The main contributions of the research are summarized as follows.

1. Performance analysis: The developed TPNS simulation model is able to

create alternative scenarios. There are two levels of creating alternative TPNS

scenarios. The first level of creating alternative scenarios involves varying

182

TPNS configurations. This includes varying the number of pipes involved,

number of compressor stations, the number of loops and the number of

junction points within the TPNS. The second level of generating alternative

networks involves varying the diameter of the pipes, the length of the pipes,

the number of compressors working within the stations, the type of

compressors, and the range of the speed of the compressors. The former

method is used to evaluate possible TPNS configurations in order to guide for

the selection of optimal network. The later method is used for evaluating the

operation of the existing system. Hence, the developed TPNS simulation

model could be used as tool for assisting decisions in designing and operating

TPNS.

2. Addressing the non-pipe elements: Investigation on various literatures on

TPNS simulation indicated that the overall operating cost of the system is

highly dependent upon the operating cost of the compressor stations in a

network [5, 6]. Hence, the detail incorporation of all its parameters, namely:

speed, suction pressure, discharge pressure, flow rates, and suction

temperatures are essential for a complete simulation of gas networks. The

proposed TPNS simulation model incorporated compressor stations by

integrating the characteristics map of compressor and energy equation as seen

in equation (3.10). Thus, this can provide more details for the analysis of

TPNS as presented in 4.3.4 on page 132.

3. Effect of age of the pipe: As discussed in section 4.4.2 on page 150, pressure

drop and flow rate of the gas are affected as the age of the pipe increases.

However, no known studies on the relationships between the age of the pipe

and its effect on pressure drop and flow were reported in the literatures. The

ability of the proposed enhanced Newton-Raphson based TPNS simulation

model to incorporate the age of the pipes in its flow equations as presented in

section 3.5.2 contributes for analyzing the effect of the age of pipes on

183

performance of the system. This could be useful for assisting decision in

maintenance and pipe replacement during the operation of TPNS.

4. Two phase flow analysis: When liquid exists in TPNS, single phase flow

modeling approach might not be adequate to predict the pressure drop, flow

capacity, and power consumption for the system. The proposed enhanced

Newton-Raphson based TPNS simulation model consists of module to

analyze pipeline network systems with two-phase by incorporating modified

homogeneous flow equation into the governing simulation equations as

presented in 3.5.1 on page 88. This could help to predict the transport

capabilities of the TPNS transporting two-phase gas-liquid mixtures.

5.3 Recommendations

The developed TPNS simulation model is mainly focused on natural gas

transmission pipeline network system. However, the principles used in TPNS

simulation model could be easily extended to be applied for the analysis of pipeline

network systems for other petroleum products with multiple sources.

The developed TPNS simulation model is able to make performance analysis for

the networks involving flow capacity, compressor ratio and power consumption for

the system. The addition of cost evaluation module to the current TPNS simulation

model could make the simulation model to analyze and evaluate the various network

configurations and operational scenarios based on the cost.

Pipeline network system mainly consists of pipes and many other non-pipe

devices such as compressor stations, valves and regulators. Although compressor

station is the key characteristics in the network, the TPNS simulation model takes

into account other non-pipe elements such as valves, regulators, and scrubbers could

be one area of research which needs further investigation.

184

In a pipeline network system problem, the system can be modeled as steady state

or transient model depending on how the gas flow changes with respect to time.

Transient analysis requires the use of partial differential equations to describe the

relationships between parameters. In case of transient simulation, variables of the

system, such as pressures and flows, are function of time. Transient TPNS simulation

model could be one of the problems to be addressed from the simulation perspective.

185

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APPENDIX A

SNAPSHOT OF THE DEVELOPED TPNS SIMULATION CODE USING

VISUAL C++

SNAPSHOT OF THE DEVELOPED TPNS SIMULATION CODE USING

VISUAL C++

APPENDIX B

RESULTS OF TPNS SIMULATION MODEL FOR GUNBARREL NETWORK

RESULTS OF TPNS SIMULATION MODEL FOR GUNBARREL NETWORK

*** W E L C O M E T O T P N S S I M U L A T I O N ***** This Program Analyses a TPNS with 3 pipes and two CSs serving single customer** *** P I P E L I N E D E T A I L S ** Input the Length of the pipes [km] 80 80 80 Input the diameter of the pipes[mm] 900 900 900 Input the service life of the pipes [year] 0 0 0 *** P I P E L I N E F R I C T I O N F A C T O R ** 0.00700345 0.00700345 0.00700345 *** P I P E L I N E F L O W C O N S T A N T S ** 7.21113e‐006 7.21113e‐006 7.21113e‐006 *** C O M P R E S S O R S T A T I O N D E T A I L S ** Input the number of the compressors within the station 1: 1 Input the number of the compressors within the station 2:1 Input speeds of the compressors 8000 8000 *** C U S T O M E R S P E C I F I CA T I O N S ** Input demand pressure requirements at customer each station PD1 =4000 Enter the initial estimates for all unknowns: **Note that the first 4 elements are pressure parameters and the next element is flow parameters** Enter the initial estimation for the unknown parameters:4000 Enter the number of iterations to go:10 THE SOLUTION IN 2 iteration(s) x_new[0] = 3103.27 x_new[1] = 3573.37 x_new[2] = 3551.63 x_new[3] = 4021.73 x_new[4] = 3.01569e+006 The tolerance in 2 iterations =0.998674 THE SOLUTION IN 3 iteration(s) x_new[0] = 383.496 x_new[1] = 3994.87 x_new[2] = 1709.82 x_new[3] = 6020.34 x_new[4] = 1.88147e+006 The tolerance in 3 iterations =0.602838 THE SOLUTION IN 4 iteration(s) x_new[0] = 976.174 x_new[1] = 3902.12 x_new[2] = 2849.15 x_new[3] = 5036.5 x_new[4] = 1.25024e+006

The tolerance in 4 iterations =0.504891THE SOLUTION IN 5 iteration(s) x_new[0] = 2062.99 x_new[1] = 4313.44 x_new[2] = 3620.2 x_new[3] = 4694.88 x_new[4] = 953728 The tolerance in 5 iterations =0.310895.THE SOLUTION IN 6 iteration(s) x_new[0] = 2450.26 x_new[1] = 3550.52 x_new[2] = 3036.26 x_new[3] = 4386.49 x_new[4] = 705598 The tolerance in 6 iterations =0.351659. THE SOLUTION IN 7 iteration(s) x_new[0] = 2471.58 x_new[1] = 3505.17 x_new[2] = 3064.83 x_new[3] = 4346.65 x_new[4] = 636963 The tolerance in 7 iterations =0.107755. THE SOLUTION IN 8 iteration(s) x_new[0] = 2472.77 x_new[1] = 3505.01 x_new[2] = 3065.89 x_new[3] = 4345.74 x_new[4] = 632577 The tolerance in 8 iterations =0.00693339. THE SOLUTION IN 9 iteration(s) x_new[0] = 2472.77 x_new[1] = 3505.01 x_new[2] = 3065.9 x_new[3] = 4345.73 x_new[4] = 632559 The tolerance in 9 iterations =2.78791e‐005 x_new[0] = 2472.77 x_new[1] = 3505.01 x_new[2] = 3065.9 x_new[3] = 4345.73 x_new[4] = 632559 The tolerance in 10 iterations =4.48679e‐010 THE SOLUTION IN 11 iteration(s) x_new[0] = 2472.77 x_new[1] = 3505.01 x_new[2] = 3065.9 x_new[3] = 4345.73 x_new[4] = 632559 The tolerance in 11 iterations =1.02997e‐015 The solution within the specified number of iteration is : x_new[0] = 2472.77 x_new[1] = 3505.01 x_new[2] = 3065.9 x_new[3] = 4345.73 x_new[4] = 632559 The compression ratio for the first station is = 1.41744 The compression ratio for the second station is = 1.41744 The energy consumption in KW for the system is = 15855.6 : Press any key to continue

APPENDIX C

SUCCESSIVE SUBSTITUTION BASED SIMULATION RESULTS (CASE 1A)

SUCCESSIVE SUBSTITUTION BASED SIMULATION RESULTS (CASE 1A)

Intr. No P1 P2 P3 Q1 QC2 QC1 0 - - 6000.00 100.00 - - 1 3678.87 6000.04 3666.79 6291.69 12899.49 -12799.49 2 2434.75 3910.86 1814.21 12715.58 7199.71 -908.02 3 2216.57 3293.23 3538.23 13340.78 510.76 12204.81 4 3106.64 4564.45 2419.12 10095.38 6865.21 6475.57 5 2107.34 3257.87 2415.34 13622.00 3642.51 6452.87 6 2605.59 3808.31 3075.67 12159.87 3629.74 9992.26 7 2697.67 4045.72 2429.77 11833.15 5620.64 6539.23 8 2339.99 3527.85 2717.10 12998.04 3678.31 8154.84 9 2643.73 3908.45 2765.30 12026.99 4587.10 8410.94 10 2525.27 3795.38 2560.44 12428.92 4731.16 7295.84 11 2485.71 3711.21 2749.06 12556.30 4103.91 8325.01 12 2593.01 3863.06 2664.90 12202.98 4682.82 7873.48 13 2501.94 3749.52 2646.67 12504.45 4428.83 7774.15 14 2536.94 3782.87 2712.73 12390.72 4372.96 8131.49 15 2551.62 3812.07 2654.48 12342.23 4573.96 7816.76 16 2517.00 3763.40 2678.15 12455.83 4396.93 7945.30 17 2544.66 3797.50 2685.79 12365.29 4469.23 7986.60 18 2535.14 3789.06 2664.78 12396.63 4492.46 7872.83 19 2529.74 3779.00 2682.37 12414.33 4428.47 7968.16 20 2540.62 3794.13 2675.73 12378.60 4482.09 7932.24 21 2532.30 3783.98 2672.87 12405.94 4461.89 7916.71 22 2534.91 3786.13 2679.53 12397.39 4453.15 7952.79 23 2536.82 3789.53 2674.20 12391.11 4473.44 7923.95 24 2533.39 3784.81 2676.04 12402.37 4457.22 7933.89 25 2535.89 3787.83 2677.08 12394.17 4462.81 7939.56 26 2535.20 3787.31 2674.98 12396.44 4466.00 7928.16 27 2534.53 3786.17 2676.58 12398.63 4459.59 7936.85 28 2535.61 3787.64 2676.08 12395.10 4464.48 7934.15 29 2534.86 3786.75 2675.71 12397.55 4462.96 7932.14 30 2535.04 3786.86 2676.38 12396.96 4461.83 7935.73 31 2535.27 3787.24 2675.89 12396.21 4463.85 7933.12 32 2534.93 3786.78 2676.03 12397.32 4462.38 7933.84 33 2535.15 3787.05 2676.16 12396.59 4462.78 7934.54 34 2535.11 3787.03 2675.95 12396.73 4463.18 7933.41 35 2535.03 3786.90 2676.09 12396.99 4462.54 7934.19

36 2535.14 3787.04 2676.06 12396.64 4462.98 7934.00 37 2535.07 3786.96 2676.01 12396.86 4462.88 7933.77 38 2535.08 3786.97 2676.08 12396.83 4462.74 7934.12 39 2535.11 3787.01 2676.04 12396.74 4462.94 7933.88 40 2535.07 3786.96 2676.04 12396.85 4462.81 7933.93 41 2535.09 3786.99 2676.06 12396.78 4462.84 7934.01 42 2535.09 3786.99 2676.04 12396.79 4462.88 7933.90 43 2535.08 3786.97 2676.05 12396.82 4462.82 7933.97 44 2535.09 3786.99 2676.05 12396.79 4462.86 7933.96 45 2535.09 3786.98 2676.04 12396.80 4462.85 7933.93 46 2535.09 3786.98 2676.05 12396.80 4462.84 7933.97 47 2535.09 3786.98 2676.05 12396.79 4462.86 7933.95 48 2535.09 3786.98 2676.05 12396.80 4462.85 7933.95 49 2535.09 3786.98 2676.05 12396.80 4462.85 7933.96 50 2535.09 3786.98 2676.05 12396.80 4462.85 7933.95

APPENDIX D

DIVERGENT CASE IN TPNS SIMULATION (CASE 2C)

DIVERGENT CASE IN TPNS SIMULATION (CASE 2C)

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